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In this study, the assumption of being differentiable for the convex function f in the (p, q)-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of (p, q)-Hermite-Hadamard inequality is proved. By using established identity, some (p, q)-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.

Some inequalities of Hermite-Hadamard type for GA-convex functions de…ned on positive intervals are given. 1. Introduction Let I (0;1) be an interval; a real-valued function f : I ! R is said to be GA-convex (concave) on I if (1.1) f x y ( ) (1 ) f (x) + f (y) for all x; y 2 I and 2 [0; 1]. Since the condition (1.1) can be written as (1.2) f exp ((1 ) lnx+ ln y) ( ) (1 ) f exp (lnx) + f exp (ln y) ; then we observe that f : I ! R is GA-convex (concave) on I if and only if f exp is convex (concave) on ln I := fln z; z 2 Ig : If I = [a; b] then ln I = [ln a; ln b] : It is known that the function f (x) = ln (1 + x) is GA-convex on (0;1) [4]. For real and positive values of x, the Euler gamma function and its logarithmic derivative , the so-called digamma function, are de…ned by (x) := Z 1 0 t e dt and (x) := 0 (x) (x) : It has been shown in [54] that the function f : (0;1)! R de…ned by f (x) = (x) + 1 2x is GA-concave on (0;1) while the function g : (0;1)! R de…ned by g (x) = (x) + 1 2x + 1 12x2 is GA-convex on (0;1) : If [a; b] (0;1) and the function g : [ln a; ln b] ! R is convex (concave) on [ln a; ln b] ; then the function f : [a; b] ! R, f (t) = g (ln t) is GA-convex (concave) on [a; b] : Indeed, if x; y 2 [a; b] and 2 [0; 1] ; then f x y = g ln x y = g [(1 ) lnx+ ln y] ( ) (1 ) g (lnx) + g (ln y) = (1 ) f (x) + f (y) showing that f is GA-convex (concave) on [a; b] : 1991 Mathematics Subject Classi…cation. 26D15; 25D10. Key words and phrases. Convex functions, Integral inequalities, GA-Convex functions. 1 2 S. S. DRAGOMIR We recall that the classical Hermite-Hadamard inequality that states that (1.3) f a+ b 2 1 b a Z b a f (t) dt f (a) + f (b) 2 for any convex function f : [a; b]! R. For related results, see [1]-[20], [23]-[25], [26]-[35] and [36]-[46]. In [54] the authors obtained the following Hermite-Hadamard type inequality. Theorem 1. If b > a > 0 and f : [a; b]! R is a di¤erentiable GA-convex (concave) function on [a; b] ; then (1.4) f (I (a; b)) ( ) 1 b a Z b a f (t) dt ( ) b L (a; b) b a f (b)+ L (a; b) a b a f (a) : The identric mean I (a; b) is de…ned by I (a; b) := 1 e b aa 1 b a while the logarithmic mean is de…ned by L (a; b) := b a ln b ln a The di¤erentiability of the function is not necessary in Theorem 1 for the …rst inequality (1.4) to hold. A proof of this fact is proved below after some short preliminaries. The second inequality in (1.4) has been proved in [54] without differentiability assumption. 2. Preliminaries We recall some facts on the lateral derivatives of a convex function. Suppose that I is an interval of real numbers with interior I and f : I ! R is a convex function on I. Then f is continuous on I and has …nite left and right derivatives at each point of I. Moreover, if x; y 2 I and x < y; then f 0 (x) f 0 + (x) f 0 (y) f 0 + (y) which shows that both f 0 and f 0 + are nondecreasing function on I. It is also known that a convex function must be di¤erentiable except for at most countably many points. For a convex function f : I ! R, the subdi¤erential of f denoted by @f is the set of all functions ' : I ! [ 1;1] such that ' °I R and f (x) f (a) + (x a)' (a) for any x; a 2 I: It is also well known that if f is convex on I; then @f is nonempty, f 0 , f 0 + 2 @f and if ' 2 @f , then f 0 (x) ' (x) f 0 + (x) for any x 2 I. In particular, ' is a nondecreasing function. If f is di¤erentiable and convex on I, then @f = ff 0g : Now, since f exp is convex on [ln a; ln b] it follows that f has …nite lateral derivatives on (ln a; ln b) and by gradient inequality for convex functions we have (2.1) f exp (x) f exp (y) (x y)' (exp y) exp y where ' (exp y) 2 f 0 (exp y) ; f 0 + (exp y) for any x; y 2 (ln a; ln b) : INEQUALITIES OF HERMITE-HADAMARD TYPE FOR GA-CONVEX FUNCTIONS 3 If s; t 2 (a; b) and we take in (2.1) x = ln t; y = ln s; then we get (2.2) f (t) f (s) (ln t ln s)' (s) s where ' (s) 2 f 0 (s) ; f 0 + (s) : Now, if we take the integral mean on [a; b] in the inequality (2.2) we get 1 b a Z b a f (t) dt f (s) 1 b a Z b a ln tdt ln s ! ' (s) s and since 1 b a Z b a ln tdt = ln I (a; b) then we get (2.3) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s)' (s) s for any s 2 (a; b) and ' (s) 2 f 0 (s) ; f 0 + (s) : This is an inequality of interest in itself. Now, if we take in (2.3) s = I (a; b) 2 (a; b) then we get the …rst inequality in (1.4) for GA-convex functions. If f is di¤erentiable and GA-convex on (a; b) ; then we have from (2.3) the inequality (2.4) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s) f 0 (s) s for any s 2 (a; b) : If we take in (2.4) s = a+b 2 = A (a; b) ; then we get (2.5) 1 b a Z b a f (t) dt f (A (a; b)) f 0 (A (a; b))A (a; b) ln A (a; b) I (a; b) : If we assume that f 0 (A (a; b)) 0; then, since I (a; b) A (a; b) ; we get (2.6) 1 b a Z b a f (t) dt f (A (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Also, if we take in (2.4) s = L (a; b) ; then we get (2.7) 1 b a Z b a f (t) dt f (L (a; b)) + f 0 (L (a; b))L (a; b) ln I (a; b) L (a; b) : If we assume that f 0 (L (a; b)) 0; then we get from (2.7) that (2.8) 1 b a Z b a f (t) dt f (L (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Now, if we take in (2.4) s = p ab = G (a; b) ; then we get (2.9) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) ln I (a; b) G (a; b) : 4 S. S. DRAGOMIR Since ln I (a; b) G (a; b) = ln I (a; b) lnG (a; b) = b ln b a ln a b a 1 ln a+ ln b 2 = a+ b 2 ln b ln a b a 1 = A (a; b) L (a; b) L (a; b) ; then (2.9) is equivalent to (2.10) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) A (a; b) L (a; b) L (a; b) : If f 0 (G (a; b)) 0; then we have (2.11) 1 b a Z b a f (t) dt f (G (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Motivated by the above results we establish in this paper other inequalities of Hermite-Hadamard type for GA-convex functions. Applications for special means are also provided. 3. New Results We start with the following result that provide in the right side of (1.4) a bound in terms of the identric mean. Theorem 2. Let f : [a; b] (0;1) ! R be a GA-convex (concave) function on [a; b] : Then we have 1 b a Z b a f (t) dt ( ) (ln b ln I (a; b)) f (a) + (ln I (a; b) ln a) f (b) ln b ln a (3.1) = b L (a; b) b a f (b) + L (a; b) a b a f (a) : Proof. Since is a GA-convex (concave) function on [a; b] then f exp is convex (concave) and we have f (t) = f exp (ln t) = f exp (ln b ln t) ln a+ (ln t ln a) ln b ln b ln a (3.2) ( ) (ln b ln t) f exp (ln a) + (ln t ln a) f exp (ln b) ln b ln a = (ln b ln t) f (a) + (ln t ln a) f (b) ln b ln a for any t 2 [a; b] : This inequality is of interest in itself as well. If we take the integral mean in (3.2) we get 1 b a Z b

A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.

Some unweighted and weighted inequalities of Hermite-Hadamard type for log-convex functions defined on real intervals are given.

A characterization of h-convex function via Hermite-Hadamard inequality related to the h-convex functions is investigated. In fact it is determined that under what conditions a function is h-convex, if it satisfies the h-convex version of Hermite-Hadamard inequality.

Some trapezoid and mid-point type inequalities related to the Hermite–Hadamard inequality on the disk of center $C=(a,b)$ and radius R, $D(C,R)\subseteq \mathbb{R}^{2}$ , are investigated. It is shown that the estimated value obtained in the trapezoid and mid-point type inequalities has a relation with the integral of the partial derivative of the considered function on $\partial (C,R)$ , the boundary of $D(C,R)$ .

Some inequalities of Hermite–Hadamard type for HH-convex functions defined on positive intervals are given. Applications for special means are also provided.

In this paper we establish some inequalities of Hermite-Hadamard type for operator convex functions and positive maps. Applications for power function and logarithm are also provided.

In this paper we provide several bounds for the modulus of the \textit{%complex \v{C}eby\v{s}ev functional}%\begin{equation*}C\left( f,g\right) :=\frac{1}{b-a}\int_{a}^{b}f\left( t\right) g\left(t\right) dt-\frac{1}{b-a}\int_{a}^{b}f\left( t\right) dt\int_{a}^{b}g\left(t\right) dt\end{equation*}%under various assumptions for the integrable functions $f,$ $g:\left[ a,b%\right] \rightarrow \mathbb{C}$. We show amongst others that, if $f$ and $g$are absolutely continuous on $\left[ a,b\right] $ with $f^{\prime }\in L_{p}%\left[ a,b\right] ,$ $g^{\prime }\in L_{q}\left[ a,b\right] ,$ $p,$ $q>1$and $\frac{1}{p}+\frac{1}{q}=1$, then%\begin{equation*}\max \left\{ \left\vert C\left( f,g\right) \right\vert ,\left\vert C\left(\left\vert f\right\vert ,g\right) \right\vert ,\left\vert C\left(f,\left\vert g\right\vert \right) \right\vert ,\left\vert C\left( \left\vertf\right\vert ,\left\vert g\right\vert \right) \right\vert \right\}\end{equation*}%\begin{equation*}\leq \left[ C\left( \ell ,F_{\left\vert f^{\prime }\right\vert ^{p}}\right) %\right] ^{1/p}\left[ C\left( \ell ,F_{\left\vert g^{\prime }\right\vert^{q}}\right) \right] ^{1/q},\end{equation*}%where $F_{\left\vert h\right\vert }:\left[ a,b\right] \rightarrow \mathbb{[}%0,\infty )$ is defined by $F_{\left\vert h\right\vert }\left( t\right):=\int_{a}^{t}$.$\left\vert h\left( t\right) \right\vert dt$ and $\ell :%\left[ a,b\right] \rightarrow \left[ a,b\right] ,$ $\ell \left( t\right) =t$is the identity function on the interval $\left[ a,b\right] .$ Applicationsfor the trapezoid inequality are also provided.

In this paper, using integral representations for n-times differentiable mappings, we establish new generalizations of certain Hermite-Hadamard type inequality for convex functions by using fairly elementary analysis. Also a parallel development is made base on concavity.

In this note we show that the Wang-Zhang inequality can be naturally applied to obtain an elegant reverse for the classical Schwarz inequality in complex inner product spaces.

Some inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given.

A mapping \(M_g(t)\) is considered to obtain some preliminary results and a new trapezoidal form of Fejer inequality related to the h-convex functions. Furthermore the obtained results are applied to achieve some new inequalities in connection with special means, random variable and trapezoidal formula.

Some fundamental results for the area balance of an abosulutely continuous function are provided

In this paper we obtain new inequalities for relative operator entropy S(A|B) in the case of operators satisfying the condition mA ≤ B ≤ MA, with 0 < m < M .

In this paper, the notion of geometrically symmetric functions is introduced. A new identity involving geometrically symmetric functions is established, and by using the obtained identity, the H?lder integral inequality and the notion of geometrically-arithmetically convexity, some new Fej?r type integral inequalities are presented. Applications of our results to special means of positive real numbers are given as well.

In this paper we establish some Hermite-Hadamard type inequalities for (m, M)-Ψ-convex functions when Ψ=- ln. Applications for power functions and weighted arithmetic mean and geometric mean are also provided.

In this paper we obtain some inequalities for isotonic functionals via a reverse and refinement of Young's inequality due to Kittaneh and Manasrah.

In this paper by using the notion of sesquilinear form we introduce a new class of numerical range and numerical radius in normed space 𝒱, also its various characterizations are given. We apply our results to get some inequalities.

Some inequalities of Hermite-Hadamard type for HGconvex functions defined on positive intervals are given. Applications for special means are also provided.

Some trace inequalities of Cassels type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

Preliminaries of q -calculus for functions of two variables over finite rectangles in the plane are introduced. Some q -analogues of the famous Hermite–Hadamard inequality of functions of two variables defined on finite rectangles in the plane are presented. A q 1 q 2 -Holder inequality for functions of two variables over finite rectangles is also established to provide some quantum estimates of trapezoidal type inequality of functions of two variables whose q 1 q 2 -partial derivatives in absolute value with certain powers satisfy the criteria of convexity on co-ordinates.

By the use of some inequalities for nonnegative Hermitian forms various inequalities for sequences and power series of bounded linear operators in complex Hilbert spaces are established. Applications for some fundamental functions of interest are also given.

In this paper we establish a generalization of the left Fejer inequality for general Lebesgue integral on measurable spaces as well as various upper bounds for the di¤erence 1 R b a g (x) dx Z b a h (x) g (x) dx h a+ b 2 where h : [a; b] ! R is a convex function and g : [a; b] ! [0;1) is an integrable weight. Applications for discrete means and Hermite-Hadamard type inequalities are also provided.

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

In this paper we show that the celebrated Davis-Choi-Jensen’s inequality for normalised positive linear maps can be extended in a weakened form for convex functions. A reverse inequality and applications for important instances of convex (concave) functions are also given.

In this paper, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are s-convex functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are s-convex functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal rule and to special means of established results are given.

In this paper, we establish some Fejér-type inequalities for convex functions. They complement the results from the previous recent paper [Dragomir, S. S.—Milošević, D. S.——Sándor, J.: On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad. Publ. Elek. Fak. Sci. Math. 4 (1993), 3–10].

Some Hermite-Hadamard’s mid-point type inequalities related to Katugampola fractional integrals are obtained where the first derivative of considered mappings is Lipschitzian or convex. Also some mid-point type inequalities are given for Lipschitzian mappings, with the aim of generalizing the results presented in previous works. Finally, some generalized inequalities in connection with special means are provided.

In this note we show that the Kre¼¬n-Lin triangle inequality can be naturally applied to obtain an elegant reverse for a classical numerical radius power inequality for bounded linear operators on complex Hilbert space due to C. Pearcy.