\(h\)-Strongly \(E\)-Convex Functions

Daniela Marian\(^{\ast }\)

January 12, 2011.

\(^{\ast }\)Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Constantin Daicoviciu, no. 15, 400020 Cluj-Napoca, Romania, e-mail: daniela.marian@math.utcluj.ro.

Starting from strongly \(E\)-convex functions introduced by E. A.
Youness, and T. Emam, from \(h\)-convex functions introduced by S. Varošanec and from the more general concept of \(h\)-convex functions introduced by A. Házy we define and study \(h\)-strongly \(E\)-convex functions. We study some properties of them.

MSC. 26B25.

Keywords. Strongly \(E\)-convex sets, strongly \(E\)-convex functions, \(h\)-convex functions, \(h\)-strongly \(E\)-convex functions.

1 Preliminary notions and results

The concepts of \(E\)-convex sets and \(E\)-convex functions were introduced by Youness in [ 8 ] . Subsequently, Chen introduced a new concept of semi-\(E\)-convex functions in [ 2 ] . Based upon these approaches, in [ 9 ] Youness and Emam introduced the concepts of strongly \(E\)-convex sets and strongly \(E\)-convex functions. We firstly recall the definitions of convex sets, convex functions, \(E\)-convex sets and \(E\)-convex functions then of strongly \(E\)-convex sets and strongly \(E\)-convex functions and finally the definitions of \(h\)-convex functions, in the sense of Varošanec [ 7 ] and Házy [ 4 ] .

A set \(A\subset \mathbb {R}^{n}\, \) is called convex if \(\lambda x+\left( 1-\lambda \right) y\in A\), for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \).

A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called convex on a convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied:

\begin{equation*} f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq \lambda f(x)+\left( 1-\lambda \right) f(y). \end{equation*}

We consider a function \(E:\mathbb {R}^{n}\to \mathbb {R}^{n}\).

[ 8 ] A set \(A\subset \mathbb {R}^{n}\, \) is called \(E\)-convex if \(\lambda E(x)+\left( 1-\lambda \right) E(y)\in A\), for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] .\)

[ 8 ] A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called \(E\)-convex on an \(E\)-convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied:

\begin{equation*} f\left( \lambda E(x)+\left( 1-\lambda \right) E(y)\right) \leq \lambda f(E(x))+\left( 1-\lambda \right) f(E(y)). \end{equation*}

[ 9 ] A set \(A\subset \mathbb {R}^{n}\, \) is called strongly \(E\)-convex if

\begin{equation*} \lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) \in A, \end{equation*}

for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\).

[ 9 ] A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called strongly \(E\)-convex on a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), the following inequality is satisfied:

\begin{equation*} f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) ) \leq \lambda f(E(x))+\left( 1-\lambda \right) f(E(y)). \end{equation*}

In the following lines we recall the definition of \(h\)-convex functions introduced in [ 7 ] by S. Varošanec.

We consider I and J intervals in \(\mathbb {R}\), \(\left( 0,1\right) \subseteq J\) and the real non-negative functions \(h:J\to \mathbb {R},\) \(f:I\to \mathbb {R},h\neq 0.\)

[ 7 ] The function \(f:I\to \mathbb {R}\ \)is called \(h\)-convex on I or is said to belong to the class \(SX\left( h,I\right) \) if for every pair of points \(x,y\in I\) and every \(\lambda \in \left( 0,1\right) \), the following inequality is satisfied:

\begin{equation*} f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq h(\lambda )f(x)+h\left( 1-\lambda \right) f(y). \end{equation*}

In [ 1 ] Bombardelli and Varošanec omitted the assumption that f and h are non-negative. We recall now the definitions of \(h\)-convex functions introduced in [ 4 ] by A. Házy.

Let X be a real (complex) linear space and \(A \subset X\) nonempty, convex, open. Let \(h:[0,1]\to \mathbb {R},\) \(f:A\to \mathbb {R}\).

[ 4 ] The function \(f:A\to \mathbb {R}\ \)is called \(h\)-convex on A if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied:

\begin{equation*} f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq h(\lambda )f(x)+h\left( 1-\lambda \right) f(y). \end{equation*}

2 Properties of \(\lowercase {h}\)-Strongly \(E\)-convex Functions

Starting from strongly \(E\)-convex functions and from \(h\)-convex functions in the sense of Házy we define and study \(h\)-strongly \(E\)-convex functions.

In the following lines we consider a map \(E:\mathbb {R}^{n}\to \mathbb {R}^{n}\) and a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n}.\) We also consider the functions \(h:[0,1]\to \mathbb {R},\) \(f:A\to \mathbb {R}\).

A function \(f:A\to \mathbb {R}\) is called \(h\)-strongly \(E\)-convex on \(A\) if for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), the following inequality is satisfied:

\begin{equation} f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\leq h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right). \label{hstrEcv} \end{equation}

If \(f:A\to \mathbb {R}\) is \(h\)-strongly \(E\)-convex on \(A\) and \(h(0)=0\) then

\begin{equation} f\left( \alpha x+E\left( x\right) \right) \leq h\left( 1\right) f\left( E\left( x\right) \right). \label{hstrEcv1} \end{equation}

Proof â–¼

We put \(\lambda =1\) in (1) and we obtain (2).

If the functions \(f_{i}:A\to \mathbb {R}\), \(i=1,2,\ldots ,k\) are \(h\)-strongly \(E\)-convex on \(A,\) then, for \(a_{i} \geq 0, i=1,2,\ldots ,k\) the function \(F:A\to \mathbb {R}\), \(F\left( x\right) =\overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( x\right) \) is \(h\)-strongly \(E\)-convex on \(A\).

Proof â–¼

Since the functions \(f_{i}:A\to \mathbb {R}\), \(i=1,2,\ldots ,k\) are \(h\)-strongly \(E\)-convex on \(A,\) then, for each \(x,y\in A\), every \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), we have

\begin{align*} & F(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\\ & =\overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( \lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) (\alpha y+E\left( y\right) )\right)\\ & \leq h\left( \lambda \right) \overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( E\left( x\right) \right) +h\left( 1-\lambda \right) \overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( E\left( y\right) \right) \\ & =h\left( \lambda \right) F\left( E\left( x\right) \right) +h\left( 1-\lambda \right) F\left( E\left( y\right) \right). \end{align*}

Hence the function F is \(h\)-strongly \(E\)-convex on \(A\).

We consider a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n},\) a function \(f:\mathbb {R}^{n}\to \mathbb {R}\), and a function \(\varphi :\mathbb {R}\to \mathbb {R}\) linear and nondecreasing.

If the function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is \(h\)-strongly \(E\)-convex on A then the composite function \(\varphi \circ f\) is \(h\)-strongly \(E\)-convex on A.

Proof â–¼

Since f is \(h\)-strongly \(E\)-convex on A, for each \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), we have \(f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\leq h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right)\) and hence

\begin{align*} & \left( \varphi \circ f\right)(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\\ & \leq \varphi \left[ h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right) )\right]\\ & =h\left( \lambda \right) \left( \varphi \circ f\right) \left( E\left( x\right) \right) +h\left(1- \lambda \right) \left( \varphi \circ f\right) \left( E\left( y\right) \right), \end{align*}

which implies that \(\varphi \circ f\) is \(h\)-strongly \(E\)-convex on A.

We denote \(E(x)\) by \(Ex\) for simplicity.

If the function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is non-negative and differentiable \(h\)-strongly \(E\)-convex on a strongly \(E\)-convex set A and h is a non-negative function with the property \(h\left( \lambda \right) \leq \lambda \) for every \(\lambda \in \left[ 0,1\right] \) then

\begin{equation} \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \end{equation}

for every \(x,y\in A.\)

Proof â–¼

Since f is \(h\)-strongly \(E\)-convex on A,

\begin{equation*} f(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) )\leq h\left( \lambda \right) \left( f\circ E\right) \left( x\right) +h\left( 1-\lambda \right) \left( f\circ E\right) \left( y\right) \end{equation*}

for each \(x,y\in A\), \(\lambda \in \left[0,1\right] \) and \(\alpha \in \left[0,1\right] \). Since \(h\left( x\right) \leq x\) for every \(x\in \left[ 0,1\right] \) we have

\begin{align*} & f(\left( \alpha y+Ey\right) +\lambda \left[ \left( \alpha x+Ex\right) -\left( \alpha y+Ey\right) \right] )\\ & \leq \lambda \left( f\circ E\right) \left( x\right) +\left( 1-\lambda \right) \left( f\circ E\right) \left( y\right) \\ & =\left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] \end{align*}

and hence

\begin{align*} & f \left( \alpha y+Ey\right) +\lambda \left[ \left( \alpha x+Ex\right) -\left( \alpha y+Ey\right) \right] )\nabla f\left( \alpha y+Ey\right) +O\left( \lambda ^{2}\right) \\ & \leq \left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] \end{align*}

By taking \(\alpha \rightarrow 0,\) we get

\begin{align*} & f\left( Ey\right) +\lambda \left( Ex-Ey\right) \nabla f\left( Ey\right) )+O\left( \lambda ^{2}\right)\\ & \leq \left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] . \end{align*}

Dividing by \(\lambda {\gt}0\) and taking \(\lambda \rightarrow 0,\) we obtain

\begin{equation*} \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right), \end{equation*}

for each \(x,y\in A.\)

The following theorem provides a characterization of \(h\)-strongly \(E\)-convex functions with respect to the E-monotonicity of the gradient of map, similar with that obtain from \(E\)-convex functions, by Soleimani-Damaneh in [ 3 ] .

Let \(f:\mathbb {R}^{n}\to \mathbb {R}\) be differentiable. The map \(\nabla f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is called E-monotone if

\begin{equation*} \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0, \end{equation*}

for every \(x,y\in R^{n}\).

\begin{equation} \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0 \end{equation}

for every \(x,y\in A.\)

Proof â–¼

Since f is \(h\)-strongly \(E\)-convex on A, from theorem (13) we have

\begin{equation*} \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \end{equation*}

and

\begin{equation*} \left( Ey-Ex\right) \nabla \left( f\circ E\right) \left( x\right) \leq \left( f\circ E\right) \left( y\right) -\left( f\circ E\right) \left( x\right) , \end{equation*}

for every \(x,y\in A.\) Adding these two inequalities we obtain

\begin{equation*} \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0 \end{equation*}

for every \(x,y\in A.\)

Let the functions \(g_{i}:\mathbb {R}^{n}\to \mathbb {R}\), \(i=1,2,\ldots ,m\) be \(h\)-strongly \(E\)-convex on \(\mathbb {R}^{n}\). We consider the set

\begin{equation*} M=\left\{ x\in R^{n}\mid g_{i}\left( x\right) \leq 0,i=1,2,\ldots m\right\} . \end{equation*}

If \(E\left( M\right) \subseteq M\) and the function h is positively then the set M is strongly \(E\)-convex.

Proof â–¼

Since the functions \(g_{i}:\mathbb {R}^{n}\to \mathbb {R}\), \(i=1,2,\ldots ,m\) are \(h\)-strongly \(E\)-convex on \(\mathbb {R}^{n}\) then, for every \(x,y\in M\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\) we have

\begin{align*} & g_{i}(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) )\\ & \leq h\left( \lambda \right) \left( g_{i}\circ E\right) \left( x\right) +h\left( 1-\lambda \right) \left( g_{i}\circ E\right) \left( y\right) \leq 0, \end{align*}

and hence \(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) \in M\).

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