Return to Article Details A separation of some Seiffert-type means by power means

A separation of some Seiffert-type means by power means

Iulia Costin $^{*}$ Gheorghe Toader $^{§}$

June 12, 2012

$^{*}$ Department of Computer Science, Technical University of Cluj-Napoca, Baritiu st. no. 28, Cluj-Napoca, Romania, e-mail: Iulia.Costin@cs.utcluj.ro.

$^{§}$ Department of Mathematics, Technical University of Cluj-Napoca, Baritiu st. no. 25, Cluj-Napoca, Romania, e-mail: Gheorghe.Toader@math.utcluj.ro.

Consider the identric mean $I$ , the logarithmic mean $L,$ two trigonometric means defined by H. J. Seiffert and denoted by $P$ and $T,$ and the hyperbolic mean $M$ defined by E. Neuman and J. Sándor $.$ There are a number of known inequalities between these means and some power means $A_{p} .$ We add to these inequalities some new results obtaining the following chain of inequalities

A_{0} < L < A_{1 / 3} < {P < A}_{2 / 3} < I < A_{3 / 3} < M < A_{4 / 3} < T < A_{5 / 3} .

MSC. 26E60

Keywords. Seiffert type means; power means; logarithmic mean; identric mean; inequalities of means.

1 Introduction

A mean is a function $M : R_{+}^{2} \to R_{+},$ with the property

min (a, b) \leq M (a, b) \leq max (a, b), \forall a, b > 0.

Each mean is reflexive, that is

M (a, a) = a, \forall a > 0.

This is also used as the definition of $M (a, a)$ .

A mean is symmetric if

M (b, a) = M (a, b), \forall a, b > 0;

it is homogeneous (of degree 1) if

M (t a, t b) = t \cdot M (a, b), \forall a, b, t > 0.

We shall refer here to the following symmetric and homogeneous means:

- the power means $A_{p}$ , defined by

A_{p} (a, b) = {[\frac{a^{p} + b^{p}}{2}]}^{\frac{1}{p}}, p \neq 0;

- the geometric mean $G$ , defined as $G (a, b) = \sqrt{a b},$ but verifying also the property

lim_{p \to 0} A_{p} (a, b) = A_{0} (a, b) = G (a, b);

- the identric mean $I$ defined by

I (a, b) = \frac{1}{e} {(\frac{a^{a}}{b^{b}})}^{\frac{1}{a - b}}, a \neq b;

- the Gini mean $S$ defined by

S (a, b) = {(a^{a} b^{b})}^{\frac{1}{a + b}};

- the first Seiffert mean $P$ , defined in [ 9 ] by

P (a, b) = \frac{a - b}{2 \sin^{- 1} \frac{a - b}{a + b}}, a \neq b;

- the second Seiffert mean $T$ , defined in [ 10 ] by

T (a, b) = \frac{a - b}{2 \tan^{- 1} \frac{a - b}{a + b}}, a \neq b;

- the Neuman-Sándor mean $M$ , defined in [ 6 ] by

M (a, b) = \frac{a - b}{2 \sinh^{- 1} \frac{a - b}{a + b}}, a \neq b;

- the logarithmic mean $L$ defined by

L (a, b) = \frac{a - b}{\ln a - \ln b}, a \neq b .

As remarked B.C. Carlson in [ 1 ] , the logarithmic mean can be represented also by

L (a, b) = \frac{a - b}{2 \tanh^{- 1} \frac{a - b}{a + b}}, a \neq b,

thus the last four means are very similar.

Being rather complicated, these means were evaluated by simpler means, first of all by power means. For two means $M$ and $N$ we write $M < N$ if $M (a, b) < N (a, b)$ for $a \neq b .$ It is known that the family of power means is an increasing family of means, thus

A_{p} < A_{q} if p < q .

The evaluation of a given mean $M$ by power means assumes the determination of some real indices $p$ and $q$ such that $A_{p} < M < A_{q} .$ The evaluation is optimal if $p$ is the the greatest and $q$ is the smallest index with this property. This means that $M$ cannot be compared with $A_{r}$ if $p < r < q .$

Optimal evaluation were given for the logarithmic mean in [ 5 ]

A_{0} < L < A_{1 / 3},

for the identric mean in [ 8 ]

A_{2 / 3} < I < A_{\ln 2},

and for the first Seiffert mean in [ 3 ]

A_{\ln 2 / \ln π} < P < A_{2 / 3} .

Following evaluations are also known:

A_{1 / 3} < P < A_{2 / 3},

proved in [ 4 ] ,

A_{1} < T < A_{2},

given in [ 10 ] ,

A_{1} < M < T,

as it was shown in [ 6 ] and

{S > A}_{2}

as it is proved in [ 7 ] . In [ 2 ] it is proven that

M < A_{3 / 2} < T

and using some of the above results, it is obtained the following chain of inequalities

A_{0} < L < A_{1 / 2} < P < A_{1} < M < A_{3 / 2} < {T < A}_{2} .

Here we retain another chain of inequalities

A_{0} < L < A_{1 / 3} < P < A_{2 / 3} < I < A_{1} < M < {T < A}_{2} < S .

Our aim is to prove that $A_{4 / 3}$ can be put between $M$ and $T$ and $A_{2}$ can be replaced by $A_{5 / 3} .$ We obtain so another nice separation of these means by “equidistant" power means.

2 Main results

We add to the inequalities (2) the next results.

Theorem 1

The following inequalities

{M < A}_{4 / 3} < {T < A}_{5 / 3}

hold.

Proof â–¼

As the means are symmetric and homogenous, for the first inequality

\frac{a - b}{2 \sinh^{- 1} \frac{a - b}{a + b}} < {(\frac{a^{4 / 3} + b^{4 / 3}}{2})}^{\frac{3}{4}}, a \neq b,

we can assume that $a > b$ and denote $a / b = t^{3} > 1.$ The inequality becomes

\frac{t^{3} - 1}{2 \sinh^{- 1} \frac{t^{3} - 1}{t^{3} + 1}} < {(\frac{t^{4} + 1}{2})}^{\frac{3}{4}}, t > 1,

\frac{2^{\frac{3}{4}} (t^{3} - 1)}{2 {(t^{4} + 1)}^{\frac{3}{4}}} < \sinh^{- 1} \frac{t^{3} - 1}{t^{3} + 1}, t > 1.

Denoting

f (t) = \sinh^{- 1} \frac{t^{3} - 1}{t^{3} + 1} - 2^{- \frac{1}{4}} (t^{3} - 1) {(t^{4} + 1)}^{- \frac{3}{4}}

we have to prove that $f (t) > 0$ for $t > 1.$ As $f (1) = 0,$ we want to prove that $f^{'} (t) > 0$ for $t > 1.$ We have

\begin{aligned} f^{'} (t) & = \frac{6 t^{2}}{(t^{3} + 1) \sqrt{2 (t^{6} + 1)}} - 2^{- \frac{1}{4}} \frac{3 t^{2} (t + 1)}{{(t^{4} + 1)}^{\frac{7}{4}}} \\ = \frac{3 t^{2} [2^{\frac{3}{4}} {(t^{4} + 1)}^{\frac{7}{4}} - (t + 1) (t^{3} + 1) \sqrt{t^{6} + 1}]}{2^{\frac{1}{4}} (t^{3} + 1) \sqrt{t^{6} + 1} {(t^{4} + 1)}^{\frac{7}{4}}} \end{aligned}

and so it is positive if

g (t) = {[2^{\frac{3}{4}} {(t^{4} + 1)}^{\frac{7}{4}}]}^{4} - {[(t + 1) (t^{3} + 1) \sqrt{t^{6} + 1}]}^{4}

is positive. Or

\begin{aligned} g (t) & = {(t - 1)}^{4} (7 t^{24} + 24 t^{23} + 48 t^{22} + 68 t^{21} + 112 t^{20} \\ + 184 t^{19} + 264 t^{18} + 296 t^{17} + 344 t^{16} + 428 t^{15} \\ + 512 t^{14} + 488 t^{13} + 466 t^{12} + 488 t^{11} + 512 t^{10} \\ + 428 t^{9} + 344 t^{8} + 296 t^{7} + 184 t^{5} + 112 t^{4} \\ + 68 t^{3} + 48 t^{2} + 24 t + 7) \end{aligned}

so that the property is certainly true. The second inequality is a simple consequence of (1) because $A_{4 / 3} < A_{3 / 2} .$ For the last inequality

\frac{a - b}{2 \tan^{- 1} \frac{a - b}{a + b}} < {(\frac{a^{5 / 3} + b^{5 / 3}}{2})}^{\frac{3}{5}}, a \neq b,

we can again consider $\frac{a}{b} = t^{3} > 1$ and we have to prove that

\frac{t^{3} - 1}{2 \tan^{- 1} \frac{t^{3} - 1}{t^{3} + 1}} < {(\frac{t^{5} + 1}{2})}^{\frac{3}{5}}, t > 1.

This is equivalent with the condition that the function

h (t) = \tan^{- 1} \frac{t^{3} - 1}{t^{3} + 1} - \frac{t^{3} - 1}{2^{\frac{2}{5}} {(t^{5} + 1)}^{\frac{3}{5}}}

is positive for $t > 1.$ As $h (1) = 0$ and

\begin{aligned} h^{'} (t) & = \frac{3 t^{2}}{t^{6} + 1} - \frac{3 t^{2} (t^{2} + 1)}{2^{\frac{2}{5}} {(t^{5} + 1)}^{\frac{8}{5}}} \\ = \frac{3 t^{2} [2^{\frac{2}{5}} {(t^{5} + 1)}^{\frac{8}{5}} - (t^{2} + 1) (t^{6} + 1)]}{2^{\frac{2}{5}} {(t^{5} + 1)}^{\frac{8}{5}} (t^{6} + 1)}, \end{aligned}

we have $h (t) > 0$ for $t > 1$ if $h^{'} (t) > 0$ for $t > 1,$ thus if the function

k (t) = {[2^{\frac{2}{5}} {(t^{5} + 1)}^{\frac{8}{5}}]}^{5} - {[(t^{2} + 1) (t^{6} + 1)]}^{5}

is positive for $t > 1.$ Or this is obvious because

\begin{aligned} k (t) & = {(t - 1)}^{4} (185 t^{28} + 200 t^{27} + 221 t^{26} + 365 t^{24} \\ + 410 t^{22} + 520 t^{19} + 580 t^{18} + 520 t^{17} + 430 t^{16} \\ + 400 t^{15} + 410 t^{14} + 440 t^{13} + 365 t^{12} + 284 t^{11} \\ + 221 t^{10} + 200 t^{9} + 185 t^{8} + 140 t^{7} + 90 t^{6} \\ + 60 t^{5} + 45 t^{4} + 25 t^{2} + 40 t^{3} + 12 t + 3) . \end{aligned}

Remark 2

For the factorization of the polynomials $g$ and $k$ we have used the computer algebra Maple.â–¡

Remark 3

It is an open problem for us to find a mean $N$ , related to the above mentioned means, with the property that

A_{5 / 3} < N < A_{2} .

For instance, the mean $S,$ which is similar to $I,$ is not convenient as follows from (2).â–¡

Corollary 4

For each $x \in (0, 1)$ we have the following evaluations

1 < \frac{x}{\sinh^{- 1} x} < A_{4 / 3} (1 - x, 1 + x) < \frac{x}{\tan^{- 1} x} < A_{5 / 3} (1 - x, 1 + x) .

Bibliography

1: B.C. Carlson, The logarithmic mean, Amer. Math. Monthly, 79 (1972), pp. 615–618.
2: I. Costin and G. Toader, A nice evaluation of some Seiffert type means by power means, Int. J. Math. Math. Sc., 2012, Article ID 430692, 6 pages, doi:10.1155/2012/430692.
3: P.A. Hästö, Optimal inequalities between Seiffert’s means and power means, Math. Inequal. Appl., 7 (2004) no. 1, pp. 47–53.
4: A.A. Jagers, Solution of problem 887, Niew Arch. Wisk. (Ser. 4), 12 (1994), pp. 230–231.
5: T.P. Lin, The power and the logarithmic mean, Amer. Math. Monthly, 81 (1974), pp. 879–883.
6: E. Neuman and J. Sándor, On the Schwab-Borchardt mean, Math. Panon., 14 (2003) no. 2, pp. 253–266.
7: E. Neuman and J. Sándor, Comparison inequalities for certain bivariate means, Appl. Anal. Discrete Math., 3 (2009), pp. 46–51.
8: A.O. Pittenger, Inequalities between arithmetic and logarithmic means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 678-715 (1980), pp. 15–18.
9: H.-J. Seiffert, Problem 887, Niew Arch. Wisk. (Ser. 4), 11 (1993), pp. 176–176.
10: H.-J. Seiffert, Aufgabe $β 16$ , Die Wurzel, 29 (1995), pp. 221–222.