Positive semi-definite matrices, exponential convexity for multiplicative majorization and related means of Cauchy’s type

Naveed Latif$^\ast $ Josip Pečarić$^\S $

March 1, 2010.

$^\ast $ Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan, e-mail: sincerehumtum@yahoo.com

$^\S $ University of Zagreb, Faculty Of Textile Technology Zagreb, Croatia, e-mail: pecaric@mahazu.hazu.hr

In this paper, we obtain new results concerning the generalizations of additive and multiplicative majorizations by means of exponential convexity. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results which implies exponential convexity and $\log $-convexity of these differences and also obtain Lyapunov’s and Dresher’s inequalities for these differences. We give some applications of additive and multiplicative majorizations. In addition, we introduce new means of Cauchy’s type and establish their monotonicity.

MSC. 39B62, 26A51, 26B25

Keywords. Convex function, additive majorization, multiplicative majorization, applications of majorization, positive semi-definite matrix, exponential-convexity, $\log $-convexity, Lyapunov’s inequality, Dresher’s inequality, means of Cauchy’s type.

1 Introduction and Preliminaries

For fixed $ n\geq 2$ let

\begin{equation*} \textbf{x} = (x_{1}, ..., x_{n}), \, \, \, \textbf{y} = (y_{1}, ..., y_{n} ) \end{equation*}

denote two n-tuples. Let

\begin{equation*} x_{[1]}\, \geq \, x_{[2]}\, \geq ...\geq \, x_{[n]},\, \, \, \, \, y_{[1]}\, \geq \, y_{[2]}\, \geq ...\geq \, y_{[n]}, \end{equation*}

\begin{equation*} x_{(1)}\, \leq \, x_{(2)}\, \leq ...\leq \, x_{(n)},\, \, \, \, \, y_{(1)}\, \leq \, y_{(2)}\, \leq ...\leq \, y_{(n)} \end{equation*}

be their ordered components.

Definition 1.1

(cf. [ 10 ] , p. 319) $\textbf{y}$ is said to majorize $\textbf{x}$ (or $\textbf{x}$ is said to be majorized by $\textbf{y}$), in symbol, $ \textbf{y} \succ \textbf{x} $, if

\begin{equation} \label{d1} \sum _{i=1}^m x_{[i]}\, \leq \, \sum _{i=1}^m y_{[i]} \end{equation}

1.1

holds for $ m =1, 2, ..., n-1$ and

\begin{equation} \label{d2} \sum _{i=1}^n x_{i}\, =\, \sum _{i=1}^n y_{i}. \end{equation}

1.2

Note that (1.1) is equivalent to

\begin{equation*} \sum _{i=n-m+1}^n x_{(i)}\, \leq \, \sum _{i=n-m+1}^n y_{(i)} \end{equation*}

for $ m =1, 2, ..., n-1$.

Parallel to the concept of additive majorization is the notion of multiplicative majorization (also termed log-majorization).

Definition 1.2

Let $\textbf{x}$, $\textbf{y}$ be two positive n-tuples, $\textbf{y}$ is said to be multiplicatively majorized by $\textbf{x}$, denoted by $\textbf{x} \prec _{\times } \textbf{y}$ if

\begin{equation} \label{d1.21} \prod _{i=1}^m x_{[i]}\, \leq \, \prod _{i=1}^m y_{[i]} \end{equation}

1.3

holds for $ m =1, 2, ..., n-1$ and

\begin{equation} \label{d1.22} \prod _{i=1}^n x_{i}\, =\, \prod _{i=1}^n y_{i}. \end{equation}

1.4

Note that (1.3) is equivalent to

\begin{equation*} \prod _{i=n-m+1}^n x_{(i)}\, \leq \, \prod _{i=n-m+1}^n y_{(i)} \end{equation*}

holds for $ m =1, 2, ..., n-1$.

To differentiate the two types of majorization, we sometimes use the symbol $\prec _{+}$ rather than $\prec $ to denote (additive) majorization.

The following theorem is well-known as the majorization theorem and a convenient reference for its proof is in the book of Marshall and Olkin (1979) ( [ 6 ] , p.11) (see [ 10 ] , p.320):

Theorem 1.3

Let I be an interval in $\mathbb {R}$ and $\textbf{x}$, $\textbf{y}$ be two n-tuples such that $x_{i}$, $y_{i}$ $\in I$ $(i=1, ..., n)$. Then

\begin{equation} \label{t1.201} \sum _{i=1}^n \phi (x_{i})\, \leq \, \sum _{i=1}^n \phi (y_{i}) \end{equation}

1.5

holds for every continuous convex function $ \phi :I\rightarrow \mathbb {R}$ iff $ \textbf{y} \succ $x holds.

Remark 1.4

[ 5 ] If $ \phi (x) $ is a strictly convex function then equality in (1.5) is valid iff $ x_{[i]}=y_{[i]},\, i=1,..., n $.â–¡

The following theorem can be regarded as a generalization of the majorization theorem and is proved by Fuchs (1947) in [ 4 ] (see [ 10 ] , p. 323):

Theorem 1.5

Let $\textbf{x}$, $\textbf{y}$ be two decreasing n-tuples and $\textbf{p} = (p_1, ..., p_n)$ be a real n-tuple such that

\begin{equation} \label{t1.401} \sum _{i=1}^k p_{i}\, x_{i}\, \leq \, \sum _{i=1}^k p_{i}\, y_{i}\, \, \, for \, \, \, k=1, ..., n-1; \end{equation}

1.6

and

\begin{equation} \label{t1.402} \sum _{i=1}^n p_{i}\, x_{i}\, =\, \sum _{i=1}^n p_{i}\, y_{i}. \end{equation}

1.7

Then for every continuous convex function $ \phi :I\rightarrow \mathbb {R}$, we have

\begin{equation} \label{t1.403} \sum _{i=1}^n p_{i}\, \phi (x_{i})\, \leq \, \sum _{i=1}^n p_{i}\, \phi (y_{i}). \end{equation}

1.8

Let $x(\tau ) $, $y(\tau )$ be two real-valued functions defined on an interval $ [a, b] $ such that $ \int _a^s x(\tau ){\rm d}\tau $, $ \int _a^s y(\tau ){\rm d}\tau $ both exist for all s $ \in $ $ [a, b] $.

Definition 1.6

(cf. [ 10 ] , p. 324) $y(\tau )$ is said to majorize $x(\tau )$, in symbol, $y(\tau ) \succ x(\tau ) $, for $\tau $ $ \in $ $ [a, b]$ if they are decreasing in $\tau $ $\in $ $[a, b]$ and

\begin{equation} \label{d3.1} \int _a^s x(\tau )\, {\rm d}\tau \leq \int _a^s y(\tau )\, {\rm d}\tau \, \, \, \, \, for \, \, s \in [a, b], \end{equation}

1.9

and equality in (1.9) holds for $s=b$.

The following theorem can be regarded as majorization theorem in integral case (see [ 10 ] , p. 325):

Theorem 1.7

$y(\tau ) \succ x(\tau )$ for $\tau $ $\in $ $[a , b]$ iff they are decreasing in $[a, b]$ and

\begin{equation} \label{t3.2} \int _a^b \phi (x(\tau ))\, {\rm d}\tau \, \leq \, \int _a^b \phi (y(\tau ))\, {\rm d}\tau \end{equation}

1.10

holds for every $\phi $ that is continuous and convex in $[a, b] $ such that the integrals exist.

The following theorem is a simple consequence of Theorem 12.14 in [ 11 ] (see [ 10 ] , p. 328):

Theorem 1.8

Let $x(\tau ),\, y(\tau ) : [a, b]\rightarrow \mathbb {R}$, $x(\tau )$ and $y(\tau )$ are continuous and increasing and let $G:[a, b]\rightarrow \mathbb {R}$ be a function of bounded variation.

If

\begin{equation} \label{t3.301} \int _\nu ^b x(\tau )\, {\rm d}G(\tau )\, \leq \, \int _\nu ^b y(\tau ) \, {\rm d}G(\tau )\, \, \, \, for\, \, all\, \, \nu \in [a, b], \end{equation}
1.11

and

\begin{equation} \label{t3.302} \int _a^b x(\tau )\, {\rm d}G(\tau )\, =\, \int _a^b y(\tau )\, {\rm d}G(\tau ) \end{equation}
1.12

hold then for every continuous convex function $f$, we have

\begin{equation} \label{t3.303} \int _a^b f(x(\tau ))\, {\rm d}G(\tau )\, \leq \, \int _a^b f(y(\tau ))\, {\rm d}G(\tau ). \end{equation}
1.13
If (1.11) holds then (1.13) holds for every continuous increasing convex function $f$.

Let $F(\tau )$, $G(\tau )$ be two continuous and increasing functions for $\tau \geq 0$ such that $F(0) = G(0) = 0$ and define

\begin{equation} \label{pb101} \overline{F}(\tau ) \, =\, 1 - F(\tau ),\, \, \, \, \, \, \, \, \, \, \, \, \overline{G}(\tau ) \, =\, 1 - G(\tau )\, \, \, \, \, \, for\, \, \, \, \, \, \, \tau \, \geq \, 0. \end{equation}

1.14

Definition 1.9

(cf. [ 10 ] , p. 330) $\overline{F}(\tau )$ is said to majorize $\overline{G}(\tau )$, in symbol, $\overline{F}(\tau ) \succ \overline{G}(\tau )$, for $\tau \in [0, +\infty )$ if

\begin{equation} \label{pb01} \int _0^s \overline{G}(\tau ) \, {\rm d}\tau \, \leq \, \int _0^s \overline{F}(\tau ) \, {\rm d}\tau \, \, \, \, \, \, \, \, \, \, for\, all\, \, \, s\, >\, 0, \end{equation}

1.15

and

\begin{equation} \label{pb02} \int _0^\infty \overline{G}(\tau ) \, {\rm d}\tau \, =\, \int _0^\infty \overline{F}(\tau ) \, {\rm d}\tau \, <\, \infty . \end{equation}

1.16

The following result was obtained by Boland and Proschan (1986) [ 3 ] (see [ 10 ] , p. 331):

Theorem 1.10

$\overline{F}(\tau ) \succ \overline{G}(\tau )$ for $\tau \in [0, +\infty )$ holds iff

\begin{equation} \label{pb03} \int _0^\infty \phi (\tau ) \, {\rm d}F(\tau ) \, \leq \, \int _0^\infty \phi (\tau ) \, {\rm d}G(\tau ) \end{equation}

1.17

holds for all convex functions $\phi $, provided the integrals are finite.

Definition 1.11

A function $h:(a,b)\rightarrow \mathbb {R}$ is exponentially convex function if it is continuous and

\begin{eqnarray*} \sum _{i,j=1}^{n}\xi _{i}\xi _{j}\, h\left(x_{i}\, +\, x_{j}\right)\, \geq \, 0 \end{eqnarray*}

for all $n\in \mathbb {N}$ and all choices $\xi _{i}\in \mathbb {R}$ and $x_{i}\in (a,b)$, $i=1,...,n$ such that $x_{i}+x_{j}\in (a,b)$, $1\leq i,j\leq n$.

The following proposition is given in [ 2 ] :

Proposition 1.12

Let $h:(a,b)\rightarrow \mathbb {R}$. The following propositions are equivalent.

$h$ is exponentially convex.
$h$ is continuous and

\begin{eqnarray*} \sum _{i,j=1}^{n}\xi _{i}\xi _{j}\, h\left(\tfrac {x_{i}\, +\, x_{j}}{2}\right)\, \geq \, 0, \end{eqnarray*}

for every $n\in \mathbb {N}$, every $\xi _{i}\in \mathbb {R}$ and every $x_{i},x_{j}\in (a,b)$, $1\leq i,j\leq n$.

Corollary 1.13

If $h$ is exponentially convex then

\begin{eqnarray*} det\left[h\left(\tfrac {x_{i}\, +\, x_{j}}{2}\right)\right]_{i,j=1}^{n}\, \geq \, 0, \end{eqnarray*}

for every $n\in \mathbb {N}$ and every $x_{i}\in (a,b)$, $i=1,...,n$.

Corollary 1.14

If $h:(a,b)\rightarrow \mathbb {R^{+}}$ is exponentially convex function then $h$ is a $\log $-convex function.

The following lemma is equivalent to definition of convex function (see [ 10 ] , p. 2):

Lemma 1.15

If $f$ is convex on an interval $I\subseteq \mathbb {R}$, then

\begin{equation*} f(s_1)(s_3-s_2)+f(s_2)(s_1-s_3)+f(s_3)(s_2-s_1)\geq 0, \end{equation*}

holds for every $s_1{\lt}s_2{\lt}s_3$, $s_1,s_2,s_3\in I$.

In [ 1 ] , the following result is proved:

Theorem 1.16

Let x and y be two positive $n$-tuples, y $\succ $ x,

\begin{eqnarray*} \Lambda _t\, =\, \Lambda _t\textmd{\rm (\textbf{x};\, \textbf{y})}\, :=\, \sum _{i=1}^n\varphi _t\left(y_i\right)\, -\, \sum _{i=1}^n\varphi _t\left(x_i\right), \end{eqnarray*}

and all $x_{[i]}$’s and $y_{[i]}$’s are not equal.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\Lambda _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1} \mathrm{det}\left[\Lambda _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
1.18

for $k=1,...,n.$
The function $s\rightarrow \Lambda _{s}$ is exponentially convex.
The function $s\rightarrow \Lambda _{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{1.501} \Lambda _s^{t-r}\, \le \, \Lambda _r^{t-s}\, \Lambda _t^{s-r}. \end{equation}
1.19

Similar results and corresponding Cauchy means are proved in [ 1 ] with a stronger condition that $\textbf{x}$ and $\textbf{y}$ are positive n-tuples.

In this paper we give results for generalizations of additive majorization as in [ 1 ] and multiplicative majorization. Moreover, several applications of majorization are obtained by using following important example

\begin{equation*} \big{(}\sum _{i=1}^n x_i, 0, ..., 0\big{)}\succ (x_{1}, ..., x_{n}). \end{equation*}

We also give some applications of additive and multiplicative majorizations.
In this connection, the following remark in [ 7 ] is important:
“majorization theory is the underlying mathematical theory on which the framework hings. It allows the transformation of the originally complicated matrix-valued non-convex problem into a simple scalar problem."

It was shown in [ 7 ] that additive majorization relation plays a key role in the design of linear MIMO transceivers, whereas the multiplicative majorization relation is the basis for nonlinear decision-feedback MIMO transceivers.

2 The case of non-negative sequences and functions

Lemma 2.1

Define the function

\begin{equation} \label{jn} \overline{\varphi }_{s}(x)\, :=\, \left\{ \begin{array}{ll} \tfrac {x^{s}}{s(s-1)}, & \hbox{s $\neq $ 1;} \\ x\log x, & \hbox{s = 1,} \end{array} \right. \end{equation}

2.20

where $ s \in \mathbb {R^{+}}$.
Then $\overline{\varphi }''_{s}(x)=x^{s-2}$, that is, $\overline{\varphi }_{s}(x)$ is convex for $x{\gt}0.$

In our results we use the notation $0\, {\rm log}\, 0=0$.

Theorem 2.2

Let x and y be two non-negative $n$-tuples, $\textbf{y} \succ \textbf{x}$ ,

\[ \overline{\Lambda }_t \, =\, \overline{\Lambda }_t\textmd{\rm (\textbf{x};\, \, \textbf{y})}\, :=\, \sum _{i=1}^n\, \overline{\varphi }_t(y_i) - \sum _{i=1}^n\, \overline{\varphi }_t(x_i), \]

and all $x_{[i]}$’s and $y_{[i]}$’s are not equal
Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\overline{\Lambda }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1} \mathrm{det}\left[\overline{\Lambda }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.21

for $k=1,...,n.$
The function $s\rightarrow \overline{\Lambda }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\Lambda }_{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib} \left(\overline{\Lambda _s}\right)^{t-r}\, \le \, \left(\overline{\Lambda _r}\right)^{t-s}\, \left(\overline{\Lambda _t}\right)^{s-r}. \end{equation}
2.22

Proof â–¼

(a) Consider the function

\begin{equation*} \mu (x)\, =\, \sum _{i,j}^{k}\, u_{i}u_{j}\, \varphi _{s_{ij}}(x) \end{equation*}

for $k=1,...,n$, $x{\gt}0$, $u_{i}\in \mathbb {R}$, $s_{ij}\in \mathbb {R^{+}}$, where $s_{ij}=\tfrac {s_{i}\, +\, s_{j}}{2}$ and $\varphi _{s_{ij}}$ is defined in (2.20).

We have

\begin{align*} \mu ”(x)& =\, \sum _{i,j}^{k}\, u_{i}u_{j}\, x^{{s_{ij}\, -\, 2}}=\\ & =\, \left(\sum _{i}^{k}\, u_{i}\, x^{\tfrac {s_{i}}{2}\, -\, 1}\right)^{2}\, \geq \, 0,\, \, \, x\geq 0. \end{align*}

This shows that $\mu $ is a convex function for $x\geq 0$.

Using Theorem 1.3,

\begin{eqnarray*} \sum _{m=1}^n\mu \left(y_m\right)\, -\, \sum _{m=1}^n{\mu \left(x_m\right)}\, \ge \, 0. \end{eqnarray*}

This implies

\begin{eqnarray*} \sum _{m=1}^{n}\left(\sum _{i,j}^{k}\, u_{i}u_{j}\, \overline{\varphi }_{s_{ij}}\left(y_m\right)\right)\, -\, \sum _{m=1}^{n}\left(\sum _{i,j}^{k}\, u_{i}u_{j}\, \overline{\varphi }_{s_{ij}}\left(x_m\right)\right)\, \ge \, 0, \end{eqnarray*}

or equivalently

\begin{eqnarray*} \sum _{i,j}^{k}\, u_{i}u_{j}\, \overline{\Lambda }_{s_{ij}}\, \geq \, 0. \end{eqnarray*}

From last inequality, it follows that the matrix $\left[\overline{\Lambda }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix, that is, (2.21) is valid.
(b) Note that $\overline{\Lambda }_s$ is continuous for $s \in \mathbb {R^{+}}$. Then by using Proposition 1.12, we get exponentially convexity of the function $s\rightarrow \overline{\Lambda }_{s}$.
(c) Since $\overline{\varphi }_{t}(x)$ is continuous and strictly convex function for $x{\gt}0$ and all $x_{[i]}$’s and $y_{[i]}$’s are not equal, therefore by Theorem 1.3 with $\phi =\overline{\varphi }_{t}$, we have

\begin{equation*} \sum _{i=1}^n \overline{\varphi }_{t}\left(y_{i}\right)\, {\gt}\, \sum _{i=1}^n \overline{\varphi }_{t}\left(x_{i}\right). \end{equation*}

This implies

\begin{equation*} \overline{\Lambda }_t\, =\, \overline{\Lambda }_t\textmd{\rm (\textbf{x};\, \textbf{y})}\, =\, \sum _{i=1}^n\overline{\varphi }_t\left(y_i\right)\, -\, \sum _{i=1}^n\overline{\varphi }_t\left(x_i\right)\, {\gt}\, 0, \end{equation*}

that is, $\overline{\Lambda }_t$ is positive-valued function.

A simple consequence of Corollary 1.14 is that $\overline{\Lambda }_s$ is $\log $-convex, then by definition

\begin{equation*} \log \left(\overline{\Lambda _{s}}\right)^{t-r}\, \leq \, \log \left(\overline{\Lambda _{r}}\right)^{t-s}\, +\, \log \left(\overline{\Lambda _{t}}\right)^{s-r}, \end{equation*}

which is equivalent to (2.22).

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{eq01} M_{t, s}& =\, \Big(\tfrac {\overline{\Lambda }_t}{\overline{\Lambda }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R^{+}},\, \, \, s\neq t.\\ M_{s, s}& =\, \exp \Big( \tfrac {\sum _{i=1}^n {y_i}^{s}\, \log y_i - \sum _{i=1}^n {x_i}^{s}\, \log x_i}{\sum _{i=1}^n {y_i}^{s}-\sum _{i=1}^n {x_i}^{s}} - \tfrac {2s-1}{s(s-1)}\Big) , \, \, \, s\neq 1.\nonumber \\ M_{1, 1}& =\, \exp \Big( \tfrac {\sum _{i=1}^n {y_i}\, (\log y_i)^{2} - \sum _{i=1}^n {x_i}\, (\log x_i)^{2}}{2\big(\sum _{i=1}^n {y_i}\, \log y_{i}-\sum _{i=1}^n {x_i}\, \log x_{i}\big)} - 1 \Big).\nonumber \end{align}

Theorem 2.3

Let $t, s, u, v \in \mathbb {R^{+}}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{t2.5} M_{t,s}\, \leq \, M_{u, v}. \end{equation}

2.24

Proof â–¼

Since $\overline{\Lambda }_t$ is $\log $-convex, therefore by $(\ref{eq01})$ we get $(\ref{t2.5}).$

Proof â–¼

Theorem 2.4

Let x and y be two non-negative decreasing $n$-tuples, $\textbf{p}=(p_1,...,p_n)$ be a real n-tuple and let

\[ \overline{\lambda }_t \, =\, \overline{\lambda }_t\textmd{\rm (\textbf{x},\, \textbf{y};\, \textbf{p})}\, :=\, \sum _{i=1}^np_i\, \overline{\varphi }_t(y_i) - \sum _{i=1}^np_i\, \overline{\varphi }_t(x_i), \]

such that conditions (1.6) and (1.7) are satisfied and $\overline{\lambda }_t$ is positive. Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\overline{\lambda }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \mathrm{det}\left[\overline{\lambda }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.25

for $k=1,...,n.$
The function $s\rightarrow \overline{\lambda }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\lambda }_{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \left(\overline{\lambda }_s\right)^{t-r}\, \le \, \left(\overline{\lambda }_r\right)^{t-s}\, \left(\overline{\lambda }_t\right)^{s-r}. \end{equation}
2.26

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.5 instead of Theo-
rem 1.3.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{eq02} \widetilde{M}_{t, s} \, & =\, \Big(\tfrac {\overline{\lambda }_t}{\overline{\lambda }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R^{+}},\, \, \, s\neq t.\\ \widetilde{M}_{s, s}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^n p_i\, {y_i}^{s} \, \log y_i - \sum _{i=1}^n p_i\, {x_i}^{s}\, \log x_i}{\sum _{i=1}^n p_i \, {y_i}^{s}-\sum _{i=1}^n p_i\, {x_i}^{s}} - \tfrac {2s-1}{s(s-1)}\Big) , \, \, \, s\neq 1.\nonumber \\ \widetilde{M}_{1, 1}\, & =\, \exp \Big(\tfrac {\sum _{i=1}^n p_i\, y_i\, (\log y_i)^{2}- \sum _{i=1}^n p_i\, x_i\, (\log x_i)^{2}}{2 \big( \sum _{i=1}^n p_i\, y_i\, \log y_i - \sum _{i=1}^n p_i\, x_i\, \log x_i \big)}- 1\Big).\nonumber \end{align}

Theorem 2.5

Let $t, s, u, v \in \mathbb {R^{+}}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{t2.7} \widetilde{M}_{t,s}\, \leq \, \widetilde{M}_{u, v}. \end{equation}

2.28

Proof â–¼

Since $\overline{\lambda }_t$ is $\log $-convex, therefore by $(\ref{eq02})$ we get $(\ref{t2.7}).$

Proof â–¼

Corollary 2.6

Let x be non-negative $n$-tuple and

\[ \digamma _t\, =\, \digamma _t\textmd{\rm (\textbf{x})}\, :=\, \overline{\varphi }_t\Big{(}\sum _{i=1}^n x_i\Big{)} - \sum _{i=1}^n \, \overline{\varphi }_t(x_i), \]

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\digamma _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023} \mathrm{det}\left[\digamma _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.29

for $k=1,...,n.$
The function $s\rightarrow \digamma _{s}$ is exponentially convex.
The function $s\rightarrow \digamma _{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib0123} \digamma _s^{t-r}\, \le \, \digamma _r^{t-s}\, \digamma _t^{s-r}. \end{equation}
2.30

Proof â–¼

Set ${\rm \textbf{y}}= \big{(}\sum _{i=1}^n x_i, 0, ..., 0\big{)}$ and ${\rm \textbf{x}}= (x_{1}, ..., x_{n})$ in Theorem ${\ref{th2.4}}$, we get our required results.

Proof â–¼

We define the following means of Cauchy type.

\begin{align} \label{eq03} \Delta _{t, r}(x) \, & =\, \Big(\tfrac {\digamma _t(x)}{\digamma _r(x)}\Big)^{\tfrac {1}{t-r}},\, \, \, \, \, \, \, \, t,\, r\in \mathbb {R^{+}},\, \, \, r\neq t.\\ \Delta _{r, r}(x)& =\, \exp \Big( \tfrac {\big{(} \sum _{i=1}^n x_i \big{)}^{r}\, \log \big{(}\sum _{i=1}^n x_i \big{)} - \sum _{i=1}^n {x_i}^{r}\, \log x_i}{\big{(} \sum _{i=1}^n x_i \big{)}^{r}-\sum _{i=1}^n {x_i}^{r}} - \tfrac {2r-1}{r(r-1)}\Big),\ r\neq 1.\nonumber \\ \Delta _{1, 1}(x)\, & =\, \exp \Big(\tfrac {(\sum _{i=1}^n x_i)\, (\log (\sum _{i=1}^n x_i))^{2}- \sum _{i=1}^n x_i \, (\log x_i)^{2}}{2 \big( (\sum _{i=1}^n x_i)\, (\log (\sum _{i=1}^n x_i)) - \sum _{i=1}^n x_i\, \log x_i \big)}- 1\Big).\nonumber \end{align}

Corollary 2.7

Let $t, r, u, v \in \mathbb {R^{+}}$ such that $t\leq u,$ $r\leq v$, then the following inequality is valid

\begin{equation} \label{c2.9} \Delta _{t, r}(x)\, \leq \, \Delta _{u, v}(x). \end{equation}

2.32

Proof â–¼

Since $\digamma _t(x)$ is log-convex, therefore by $(\ref{eq03})$ we get $(\ref{c2.9}).$

Proof â–¼

We define the following Cauchy means, which are similar to [ 9 ] for $p_{i}=1$, $i=1,..., n$.

\begin{equation} \label{eq04} \Upsilon ^{s}_{t, r}(x)\, =\, \Big(\tfrac {r(r-s)}{t(t-s)}.\tfrac {\big(\sum _{i=1}^n x_{i}^{s}\big)^{\tfrac {t}{s}}-\sum _{i=1}^nx_{i}^{t}}{\big(\sum _{i=1}^n x_{i}^{s}\big)^{\tfrac {r}{s}}-\sum _{i=1}^nx_{i}^{r}}\Big)^{\tfrac {1}{t-r}}, \end{equation}

2.33

$t,\, r,\, s\in \mathbb {R^{+}},\, \, t\neq r,\, \, t\neq s,\, \, r\neq s$.

\begin{equation*} \Upsilon ^{s}_{s, r}(x)\, =\, \Big( \tfrac {r(r-s)}{s^{2}}.\tfrac {\sum _{i=1}^n x_{i}^{s}\, \log \big{(}\sum _{i=1}^n x_i^{s} \big{)}-s \sum _{i=1}^nx_i^{s}\, \log x_{i}}{\big(\sum _{i=1}^n x_{i}^{s}\big)^{\tfrac {r}{s}}-\sum _{i=1}^nx_{i}^{r}}\Big)^{\tfrac {1}{s-r}} , \, \, \, r\neq s. \end{equation*}

\begin{align*} \Upsilon ^{s}_{r, r}(x)=\, \exp \Big( \tfrac {\big{(} \sum _{i=1}^n x_i^{s} \big{)}^{\tfrac {r}{s}}\, \log \big{(}\sum _{i=1}^n x_i^{s} \big{)} -s \sum _{i=1}^n {x_i}^{r}\, \log x_i}{s \Big(\big{(} \sum _{i=1}^n x_i^{s} \big{)}^{\tfrac {r}{s}}-\sum _{i=1}^n {x_i}^{r}\Big)} - \tfrac {2r-s}{r(r-s)}\Big),\\ r\neq s. \end{align*}

\begin{equation*} \Upsilon ^{s}_{s, s}(x)\, =\, \exp \Big( \tfrac {\big{(} \sum _{i=1}^n x_i^{s} \big{)}\, \big(\log \big{(}\sum _{i=1}^n x_i^{s} \big{)}\big)^{2} - s^{2} \sum _{i=1}^n {x_i}^{s}\, (\log x_i)^{2}}{2s \Big( \big{(} \sum _{i=1}^n x_i^{s} \big{)}\, \log \big( \sum _{i=1}^n x_i^{s} \big)-s \sum _{i=1}^n {x_i}^{s}\, \log x_{i}\Big)} - \tfrac {1}{s}\Big). \end{equation*}

Corollary 2.8

Let $t, r, u, v \in \mathbb {R^{+}}$ such that $t\leq u,$ $r\leq v$, then the following inequality is valid

\begin{equation} \label{c2.10} \Upsilon ^{s}_{t, r}(x)\, \leq \, \Upsilon ^{s}_{u, v}(x) \end{equation}

2.34

Proof â–¼

Let

\begin{equation} \label{tas} \digamma _{t}(x)\, :=\, \left\{ \begin{array}{ll} \tfrac {1}{t(t-1))}\Big( \Big(\sum _{i=1}^n x_i \Big)^{t} - \sum _{i=1}^n x_i^{t} \Big), & \hbox{t $\neq $ 1;} \\ \sum _{i=1}^n x_{i}\, \log \big( \sum _{i=1}^n x_{i}\big) - \sum _{i=1}^n x_{i}\, \log x_{i}, & \hbox{t = 1.} \end{array} \right. \end{equation}

2.35

Using corollary 2.7, we have

\begin{equation*} \Big( \tfrac {r(r-1)}{t(t-1)}.\tfrac {\big( \sum _{i=1}^n x_{i} \big)^{t} - \sum _{i=1}^n x_i^{t} }{\big( \sum _{i=1}^n x_{i} \big)^{r} - \sum _{i=1}^n x_i^{r}} \Big)^{\tfrac {1}{t-r}} \leq \, \Big( \tfrac {u(u-1)}{v(v-1)}.\tfrac {\big( \sum _{i=1}^n x_{i} \big)^{v} - \sum _{i=1}^n x_i^{v} }{\big( \sum _{i=1}^n x_{i} \big)^{u} - \sum _{i=1}^n x_i^{u}} \Big)^{\tfrac {1}{v-u}}. \end{equation*}

Since $s{\gt}0$ by substituting $x_{i}= x_{i}^{s}$, $t=\tfrac {t}{s}$, $r=\tfrac {r}{s}$, $u=\tfrac {u}{s}$ and $v=\tfrac {v}{s}$ in above inequality, we get

\begin{equation*} \Big( \tfrac {r(r-s)}{t(t-s)}.\tfrac {\big( \sum _{i=1}^n x_{i}^{s} \big)^{\tfrac {t}{s}} - \sum _{i=1}^n x_i^{t} }{\big( \sum _{i=1}^n x_{i}^{s} \big)^{\tfrac {r}{s}} - \sum _{i=1}^n x_i^{r}} \Big)^{\tfrac {s}{t-r}}\leq \, \Big( \tfrac {u(u-s)}{v(v-s)}.\tfrac {\big( \sum _{i=1}^n x_{i}^{s} \big)^{\tfrac {v}{s}} - \sum _{i=1}^n x_i^{v} }{\big( \sum _{i=1}^n x_{i}^{s} \big)^{\tfrac {u}{s}} - \sum _{i=1}^n x_i^{u}} \Big)^{\tfrac {s}{v-u}}. \end{equation*}

By raising power $\tfrac {1}{s}$, we get ($\ref{c2.10}$).

Proof â–¼

Remark 2.9

Let us note that in [ 8 ] , the following function $\phi _{t}= t\, \digamma _{t}$ was considered. It was proved that

\begin{equation} \label{2.111} \phi _{s}^{t-r}\, \leq \, \phi _{r}^{t-s}\, \, \phi _{t}^{s-r}. \end{equation}

2.36

In [ 9 ] , it was proved that this implies

\begin{equation*} \digamma _{s}^{t-r}\, \leq \, \tfrac {s^{t-r}}{r^{t-s}t^{s-r}}\, \, \digamma _{r}^{t-s}\, \, \digamma _{t}^{s-r}. \end{equation*}

Since $\tfrac {s^{t-r}}{r^{t-s}t^{s-r}}\, {\lt}\, 1$, we have that (2.36) is better than (2.30).â–¡

Theorem 2.10

Let $x(\tau )$ and $y(\tau ) $ be two non-negative real-valued functions defined on an interval $[a, b]$, decreasing in $[a, b]$, $y(\tau ) \succ x(\tau ) $ and

\begin{equation*} \overline{\beta }_{t}(x(\tau );\, y(\tau ))\, :=\, \int _a^b\overline{\varphi }_t(y(\tau ))\, {\rm d}\tau - \int _a^b\overline{\varphi }_t(x(\tau ))\, {\rm d}\tau , \end{equation*}

and $\overline{\beta }_{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\overline{\beta }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa10231987} \mathrm{det}\left[\overline{\beta }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.37

for $k=1,...,n.$
The function $s\rightarrow \overline{\beta }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\beta }_{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib0123798} \left(\overline{\beta }_s\right)^{t-r}\, \le \, \left(\overline{\beta }_r\right)^{t-s}\, \left(\overline{\beta }_t\right)^{s-r}. \end{equation}
2.38

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.7 instead of Theo-
rem 1.3.

Proof â–¼

Theorem 2.11

Let $ x(\tau ),\, y(\tau ) : [a, b]\rightarrow \mathbb {R}$, $x(\tau )$ and $y(\tau )$ are non-negative continuous and increasing, $G:[a, b]\rightarrow \mathbb {R}$ be a function of bounded variation and

\begin{equation*} \overline{\Gamma }_{t}(x(\tau ),\, y(\tau );\, G(\tau ))\, :=\, \int _a^b\overline{\varphi }_t(y(\tau ))\, {\rm d}G(\tau ) - \int _a^b\overline{\varphi }_t(x(\tau ))\, {\rm d}G(\tau ) \end{equation*}

such that conditions (1.11) and (1.12) are satisfied and $\overline{\Gamma }_{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\overline{\Gamma }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112} \mathrm{det}\left[\overline{\Gamma }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.39

for $k=1,...,n.$
The function $s\rightarrow \overline{\Gamma }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\Gamma }_{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312} \left(\overline{\Gamma }_s\right)^{t-r}\, \le \, \left(\overline{\Gamma }_r\right)^{t-s}\, \left(\overline{\Gamma }_t\right)^{s-r}. \end{equation}
2.40

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.8 instead of Theo-
rem 1.3.

Theorem 2.12

Let $F(\tau )$ and $G(\tau )$ are non-negative continuous and increasing functions defined on an interval $[0, +\infty )$ such that $F(0) = G(0) = 0$, $\overline{F}(\tau ) \succ \overline{G}(\tau )$, $\overline{F}(\tau )$ and $\overline{G}(\tau )$ are defined in (1.14),

\begin{equation*} \theta _{t}(\tau ,\, G(\tau );\, F(\tau ))\, :=\, \int _{0}^{\infty }\overline{\varphi }_t(\tau )\, {\rm d}G(\tau ) - \int _{0}^{\infty } \overline{\varphi }_t(\tau )\, {\rm d}F(\tau ), \end{equation*}

and $\theta _{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R^{+}}$, the matrix $\left[\theta _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112q} \mathrm{det}\left[\theta _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
2.41

for $k=1,...,n.$
The function $s\rightarrow \theta _{s}$ is exponentially convex.
The function $s\rightarrow \theta _{s}$ is a $\log $-convex on $\mathbb {R^{+}}$ and the following inequality holds for $0\, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312q} \theta _s^{t-r}\, \le \, \theta _r^{t-s}\, \theta _t^{s-r}. \end{equation}
2.42

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.10 instead of Theorem 1.3.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{jav1} \overline{\theta }_{t, s} \, & =\, \Big(\tfrac {{\theta }_t}{{\theta }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R^{+}},\, \, \, \, s\neq t.\\ \overline{\theta }_{s, s}\, & =\, \exp \Big( \tfrac {\int _{0}^{\infty } \tau ^{s}\, \log \tau \, {\rm d}G(\tau )- \int _{0}^{\infty } \tau ^{s}\, \log \tau \, {\rm d}F(\tau )}{\int _{0}^{\infty }\tau ^{s}\, {\rm d}G(\tau )- \int _{0}^{\infty }\tau ^{s}\, {\rm d}F(\tau )}-\tfrac {2s-1}{s(s-1)}\Big) , \, \, \, s\neq 1.\nonumber \\ \overline{\theta }_{1, 1}\, & =\, \exp \Big( \tfrac {\int _{0}^{\infty }\tau \, \log ^{2}\tau \, {\rm d}G(\tau )- \int _{0}^{\infty }\tau \, \log ^{2}\tau \, {\rm d}F(\tau )}{2\Big(\int _{0}^{\infty }\tau \, \log \tau \, {\rm d}G(\tau )- \int _{0}^{\infty }\tau \, \log \tau \, {\rm d}F(\tau )\Big)} - 1\Big).\nonumber \end{align}

Theorem 2.13

Let $t, s, u, v \in \mathbb {R^{+}}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{jav2} \overline{\theta }_{t,s}\, \leq \, \overline{\theta }_{u, v}. \end{equation}

2.44

Proof â–¼

Since $\theta _t$ is $\log $-convex, therefore by $(\ref{jav1})$ we get $(\ref{jav2}).$

Proof â–¼

Remark 2.14

As in [ 1 ] , we can use Theorem 2.2, Theorem 2.4, Corollary 2.6, Theorem $\ref{th3.4}$, Theorem $\ref{th3.5}$ and Theorem 2.12 to obtain corresponding Cauchy means.â–¡

3 Multiplicative Majorization

Lemma 3.1

Given $t\in \mathbb {R}$, define the function

\begin{equation} \label{l2.101} \psi _{t}(x)\, :=\, \left\{ \begin{array}{ll} \tfrac {1}{t^{2}}\, {\rm e}^{tx}, & \hbox{t\, $\neq $ 0;} \\ \tfrac {1}{2}\, x^{2}, & \hbox{t\, = 0,} \end{array} \right. \end{equation}

3.45

Then ${\psi }''_{t}(x)= {\rm e}^{tx}$, that is, $\psi _{t}(x)$ is convex for $x\in \mathbb {R}.$

Theorem 3.2

Let x and y be two real $n$-tuples, $ \textbf{y} \succ \textbf{x} $ ,

\[ {\xi }_t \, =\, {\xi }_t\textmd{\rm (\textbf{x};\, \textbf{y})}\, :=\, \sum _{i=1}^n\, {\psi }_t(y_i) - \sum _{i=1}^n\, {\psi }_t(x_i), \]

and all $x_{[i]}$’s and $y_{[i]}$’s are not equal.
Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\xi _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qz} \mathrm{det}\left[\xi _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.46

for $k=1,...,n.$
The function $s\rightarrow \xi _{s}$ is exponentially convex.
The function $s\rightarrow \xi _{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qz} \xi _s^{t-r}\, \le \, \xi _r^{t-s}\, \xi _t^{s-r}. \end{equation}
3.47

Proof â–¼

As in the proof of Theorem 2.2, we use $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{j102} \Theta _{t, s} \, & =\, \Big(\tfrac {{\xi }_t}{{\xi }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R},\, \, \, \, s\neq t.\\ \Theta _{s, s}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}y_{i}\, {\rm e}^{sy_{i}}- \sum _{i=1}^{n}x_{i}\, {\rm e}^{sx_{i}}}{\sum _{i=1}^{n}{\rm e}^{sy_{i}}-\sum _{i=1}^{n}{\rm e}^{sy_{i}}}-\tfrac {2}{s}\Big) , \, \, \, s\neq 0.\nonumber \\ \Theta _{0, 0}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}y^{3}_{i}-\sum _{i=1}^{n}x^{3}_{i}}{3\big(\sum _{i=1}^{n}y^{2}_{i}-\sum _{i=1}^{n}x^{2}_{i}\big)} \Big).\nonumber \end{align}

Theorem 3.3

Let $t, s, u, v \in \mathbb {R}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{j201} \Theta _{t,s}\, \leq \, \Theta _{u, v}. \end{equation}

3.49

Proof â–¼

Since $\xi _t$ is $\log $-convex, therefore by $(\ref{j102})$ we get $(\ref{j201}).$

Proof â–¼

Theorem 3.4

Let x and y be two decreasing real $n$-tuples, $\textbf{p}=(p_1,...,p_n)$ be a real n-tuple and let

\[ \overline{\xi }_t\, =\, \overline{\xi }_t\textmd{\rm (\textbf{x}, \textbf{y}; \textbf{p})} \, :=\, \sum _{i=1}^n\, p_i\, {\psi }_t(y_i) - \sum _{i=1}^n\, p_i\, {\psi }_t(x_i), \]

such that conditions (1.6) and (1.7) are satisfied and $\overline{\xi }_t$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\overline{\xi }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl} \mathrm{det}\left[\overline{\xi }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.50

for $k=1,...,n.$
The function $s\rightarrow \overline{\xi }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\xi }_{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl} \left(\overline{\xi }_s\right)^{t-r}\, \le \, \left(\overline{\xi }_r\right)^{t-s}\, \left(\overline{\xi }_t\right)^{s-r}. \end{equation}
3.51

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.5 instead of Theorem 1.3 and $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{j302} \overline{\Theta }_{t, s} \, & =\, \Big(\tfrac {\overline{\xi }_t}{\overline{\xi }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R},\, \, \, \, \, s\neq t.\\ \overline{\Theta }_{s, s}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}p_{i}\, y_{i}\, {\rm e}^{sy_{i}}- \sum _{i=1}^{n}p_{i}\, x_{i}\, {\rm e}{\rm e}^{sx_{i}}}{\sum _{i=1}^{n}p_{i}\, {\rm e}^{sy_{i}}-\sum _{i=1}^{n}p_{i}\, {\rm e}^{sx_{i}}}-\tfrac {2}{s}\Big) , \, \, \, s\neq 0.\nonumber \\ \overline{\Theta }_{0, 0}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}p_{i}\, y^{3}_{i}-\sum _{i=1}^{n}p_{i}\, x^{3}_{i}}{3\big(\sum _{i=1}^{n}p_{i}\, y^{2}_{i}-\sum _{i=1}^{n}p_{i}\, x^{2}_{i}\big)} \Big).\nonumber \end{align}

Theorem 3.5

Let $t, s, u, v \in \mathbb {R}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{j401} \overline{\Theta }_{t,s}\, \leq \, \overline{\Theta }_{u, v}. \end{equation}

3.53

Proof â–¼

Since $\overline{\xi }_t$ is $\log $-convex, therefore by $(\ref{j302})$ we get $(\ref{j401}).$

Proof â–¼

Corollary 3.6

Let x and y be two positive $n$-tuples, $\textbf{x} \prec _{\times } \textbf{y} $ ,

\begin{equation*} \Omega _t\textmd{\rm (log \textbf{x};\, log \textbf{y})}\, =\, \xi _t\textmd{\rm (\textbf{x};\, \textbf{y})}\, :=\, \left\{ \begin{array}{ll} \tfrac {1}{t^{2}}\, \Big(\sum _{i=1}^n y_{i}^{t} - \sum _{i=1}^n x_{i}^{t}\Big) , & \hbox{t\, $\neq $ 0;} \\ \tfrac {1}{2}\, \Big(\sum _{i=1}^n {\rm log}^{2}y_{i} - \sum _{i=1}^n {\rm log}^{2}x_{i}\Big), & \hbox{t\, = 0,} \end{array} \right. \end{equation*}

and all $x_{[i]}$’s and $y_{[i]}$’s are not equal.
Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\Omega _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl23} \mathrm{det}\left[\Omega _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.54

for $k=1,...,n.$
The function $s\rightarrow \Omega _{s}$ is exponentially convex.
The function $s\rightarrow \Omega _{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl23} \Omega _s^{t-r}\, \le \, \Omega _r^{t-s}\, \Omega _t^{s-r}. \end{equation}
3.55

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.3 for $\rm \textbf{x}=log{\rm \textbf{x}}$ and $\rm \textbf{y}=log{\rm \textbf{y}}$ and using $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{cj102} \Psi _{t, s} \, & =\, \Big(\tfrac {\Omega _t}{\Omega _s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R},\, \, \, \, \, s\neq t.\\ \Psi _{s, s}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}y_{i}^{s}\, \log y_{i} -\sum _{i=1}^{n}x_{i}^{s}\, \log x_{i}}{\sum _{i=1}^{n} y_{i}^{s}-\sum _{i=1}^{n}x_{i}^{s}}-\tfrac {2}{s}\Big) , \, \, \, \, s\neq 0.\nonumber \\ \Psi _{0, 0} \, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}\log ^{3}y_{i}-\sum _{i=1}^{n}\log ^{3}x_{i}}{3\big(\sum _{i=1}^{n}\log ^{2}y_{i}-\sum _{i=1}^{n}\log ^{2}x_{i}\big)} \Big).\nonumber \end{align}

Corollary 3.7

Let $t, s, u, v \in \mathbb {R}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{cj201} \Psi _{t,s}\, \leq \, \Psi _{u, v}. \end{equation}

3.57

Proof â–¼

Since $\Omega _t$ is $\log $-convex, therefore by $(\ref{cj102})$ we get $(\ref{cj201}).$

Proof â–¼

Corollary 3.8

Let x and y be two positive decreasing $n$-tuples, $\textbf{p} = (p_{1}, ..., p_{n} )$ be a real n-tuple and let

\begin{eqnarray*} \lefteqn{\overline{\Omega }_t\textmd{\rm (\textbf{x},\, \textbf{y};\, \textbf{p})}=} \\ & & =\, \overline{\xi }_t\textmd{\rm (log \textbf{x},\, log \textbf{y};\, \, \textbf{p})}\, :=\, \left\{ \begin{array}{ll} \tfrac {1}{t^{2}}\, \Big(\sum _{i=1}^n p_{i}\, y_{i}^{t} - \sum _{i=1}^n p_{i}\, x_{i}^{t}\Big) , & \hbox{t\, $\neq $ 0;} \\ \tfrac {1}{2}\, \Big(\sum _{i=1}^n p_{i}\, {\rm log}^{2}y_{i} - \sum _{i=1}^n p_{i}\, {\rm log}^{2}x_{i}\Big), & \hbox{t\, = 0,} \end{array} \right. \end{eqnarray*}

such that conditions (1.6) and (1.7) are satisfied and $\overline{\Omega }_t$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\overline{\Omega }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl234} \mathrm{det}\left[\overline{\Omega }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.58

for $k=1,...,n.$
The function $s\rightarrow \overline{\Omega }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\Omega }_{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl234} \left(\overline{\Omega }_s\right)^{t-r}\, \le \, \left(\overline{\Omega }_r\right)^{t-s}\, \left(\overline{\Omega }_t\right)^{s-r}. \end{equation}
3.59

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.5 for $\rm \textbf{x}=log\rm \textbf{x}$ and $\rm \textbf{y}=log\rm \textbf{y}$ and using $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{cj302} \overline{\Psi }_{t, s} \, & =\, \Big(\tfrac {\overline{\Omega }_t}{\overline{\Omega }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R},\, \, \, \, s\neq t.\\ \overline{\Psi }_{s, s}\, & =\, \exp \Big( \tfrac {\sum _{i=1}^{n}p_{i}\, y_{i}^{s}\, \log y_{i} -\sum _{i=1}^{n}p_{i}\, x_{i}^{s}\, \log x_{i}}{\sum _{i=1}^{n} p_{i}\, y_{i}^{s}-\sum _{i=1}^{n}p_{i}\, x_{i}^{s}}-\tfrac {2}{s}\Big) , \, \, \, s\neq 0.\nonumber \\ \overline{\Psi }_{0, 0} & = \exp \Big( \tfrac {\sum _{i=1}^{n}p_{i}\, \log ^{3}y_{i}-\sum _{i=1}^{n} p_{i}\, \log ^{3}x_{i}}{3\big(\sum _{i=1}^{n}p_{i}\, \log ^{2}y_{i}-\sum _{i=1}^{n}p_{i}\, \log ^{2}x_{i}\big)} \Big).\nonumber \end{align}

Corollary 3.9

Let $t,\, s,\, u,\, v \in \mathbb {R}$ such that $t\leq u,$ $s\leq v$, then the following inequality is valid

\begin{equation} \label{cj401} \overline{\Psi }_{t,s} \, \leq \, \overline{\Psi }_{u, v}. \end{equation}

3.61

Proof â–¼

Since $\overline{\Omega }_t$ is $\log $-convex, therefore by $(\ref{cj302})$ we get $(\ref{cj401}).$

Proof â–¼

Theorem 3.10

Let $x(\tau )$ and $y(\tau ) $ be two real-valued functions defined on an interval $[a, b]$, decreasing in $[a, b]$, $ y(\tau ) \succ x(\tau ) $ and

\begin{equation*} \Phi _{t}(x(\tau );\, y(\tau ))\, :=\, \int _a^b\psi _t(y(\tau ))\, {\rm d}\tau - \int _a^b\psi _t(x(\tau ))\, {\rm d}\tau , \end{equation*}

and $\Phi _{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\Phi _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl23443} \mathrm{det}\left[\Phi _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.62

for $k=1,...,n.$
The function $s\rightarrow \Phi _{s}$ is exponentially convex.
The function $s\rightarrow \Phi _{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl23443} \Phi _s^{t-r}\, \le \, \Phi _r^{t-s}\, \Phi _t^{s-r}. \end{equation}
3.63

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.7 instead of Theorem 1.3 and $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

Theorem 3.11

Let $x(\tau ),\, y(\tau ) : [a, b]\rightarrow \mathbb {R}$, $x(\tau )$ and $y(\tau )$ are continuous and increasing, $G:[a, b]\rightarrow \mathbb {R}$ be a function of bounded variation and

\begin{equation*} \overline{\Phi }_{t}(x(\tau ),\, y(\tau );\, G(\tau ))\, :=\, \int _a^b\psi _t(y(\tau ))\, {\rm d}G(\tau ) - \int _a^b\psi _t(x(\tau ))\, {\rm d}G(\tau ) \end{equation*}

such that conditions (1.11) and (1.12) are satisfied and $\overline{\Phi }_{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\overline{\Phi }_{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl23443} \mathrm{det}\left[\overline{\Phi }_{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.64

for $k=1,...,n.$
The function $s\rightarrow \overline{\Phi }_{s}$ is exponentially convex.
The function $s\rightarrow \overline{\Phi }_{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl23443} \left(\overline{\Phi }_s\right)^{t-r}\, \le \, \left(\overline{\Phi }_r\right)^{t-s}\, \left(\overline{\Phi }_t\right)^{s-r}. \end{equation}
3.65

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.8 instead of Theorem 1.3 and $\psi _{t}$ instead of $\overline{\varphi }_t$.

Proof â–¼

Theorem 3.12

Let $F(\tau )$ and $G(\tau )$ are real continuous and increasing functions defined on an interval $[0, +\infty )$ such that $F(0) = G(0) = 0$, $\overline{F}(\tau ) \succ \overline{G}(\tau )$, $\overline{F}(\tau )$ and $\overline{G}(\tau )$ are defined in (1.14),

\begin{equation*} \vartheta _{t}(\tau ,\, G(\tau );\, F(\tau ))\, :=\, \int _{0}^{\infty }\psi _t(\tau )\, {\rm d}G(\tau ) - \int _{0}^{\infty }\psi _t(\tau )\, {\rm d}F(\tau ), \end{equation*}

and $\vartheta _{t}$ is positive.

Then the following statements are valid:

For every $n\in \mathbb {N}$ and $s_{1},...,s_{n}\in \mathbb {R}$, the matrix $\left[\vartheta _{\tfrac {s_{i}+s_{j}}{2}}\right]_{i,j=1}^{n}$ is a positive semi-definite matrix. Particularly

\begin{equation} \label{nasa1023112qzl23443} \mathrm{det}\left[\vartheta _{\tfrac {s_{i}\, +\, s_{j}}{2}}\right]_{i,j=1}^{k}\, \geq \, 0 \end{equation}
3.66

for $k=1,...,n.$
The function $s\rightarrow \vartheta _{s}$ is exponentially convex.
The function $s\rightarrow \vartheta _{s}$ is a $\log $-convex on $\mathbb {R}$ and the following inequality holds for $-\infty \, {\lt}\, r\, {\lt}\, s\, {\lt}\, t\, {\lt}\, \infty :$

\begin{equation} \label{mohib012312qzl23443} \vartheta _s^{t-r}\, \le \, \vartheta _r^{t-s}\, \vartheta _t^{s-r}. \end{equation}
3.67

Proof â–¼

As in the proof of Theorem 2.2, we use Theorem 1.10 instead of Theorem 1.3 and $\psi _{t}$ instead of $\overline{\varphi }_{t}$.

Proof â–¼

As in [ 1 ] , we define the following means of Cauchy type.

\begin{align} \label{was1} \overline{\vartheta }_{t, s} \, & =\, \Big(\tfrac {{\vartheta }_t}{{\vartheta }_s}\Big)^{\tfrac {1}{t-s}},\, \, \, \, \, \, \, \, \, \, \, t,\, s\in \mathbb {R},\, \, \, \, s\neq t.\\ \overline{\vartheta }_{s, s}\, & =\, \exp \Big( \tfrac {\int _{0}^{\infty } \tau \, {\rm e}^{s\tau }\, {\rm d}G(\tau )- \int _{0}^{\infty } \tau \, {\rm e}^{s\tau }\, {\rm d}F(\tau )}{\int _{0}^{\infty }{\rm e}^{s\tau }\, {\rm d}G(\tau )- \int _{0}^{\infty }{\rm e}^{s\tau }\, {\rm d}F(\tau )}-\tfrac {2}{s}\Big) , \, \, \, s\neq 0.\nonumber \\ \overline{\vartheta }_{0, 0}\, & =\, \exp \Big( \tfrac {\int _{0}^{\infty }\tau ^{3}\, {\rm d}G(\tau )- \int _{0}^{\infty }\tau ^{3}\, {\rm d}F(\tau )}{3\big(\int _{0}^{\infty }\tau ^{2}\, {\rm d}G(\tau )- \int _{0}^{\infty }\tau ^{2}\, {\rm d}F(\tau )\big)}\Big).\nonumber \end{align}

Theorem 3.13

Let $t, s, u, v \in \mathbb {R}$ such that $t\leq u,$ $s\leq v$, then the following

\begin{equation} \label{was2} \overline{\vartheta }_{t,s}\, \leq \, \overline{\vartheta }_{u, v}. \end{equation}

3.69

Proof â–¼

Since $\vartheta _t$ is $\log $-convex, therefore by $(\ref{was1})$ we get $(\ref{was2}).$

Proof â–¼

Remark 3.14

As in [ 1 ] , we can use Theorem 3.2, Theorem 3.4, Corollary 3.6, Corollary 3.8, Theorem $\ref{jt1}$, Theorem $\ref{jt2}$ and Theorem $\ref{wasi}$ to obtain corresponding Cauchy means.â–¡

Acknowledgement

This research work is funded by Higher Education Commission Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.

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