On strongly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi_{h}}$$\end{document} -convex functions in inner product spaces

In this paper, we introduce the notion of strongly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi_{h}}$$\end{document} -convex functions with respect to c > 0 and present some properties and representation of such functions. We obtain a characterization of inner product spaces involving the notion of strongly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi_{h}}$$\end{document} -convex functions. Finally, a version of Hermite–Hadamard-type inequalities for strongly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi_{h}}$$\end{document} -convex functions is established.


Introduction
The inequalities discovered by C. Hermite and J. Hadamard for convex functions are very important in the literature (see, e.g., [7], [13, p. 137]). These inequalities state that if f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then The inequality (1) has evoked the interest of many mathematicians. Especially in the last three decades, numerous generalizations, variants and extensions of this inequality have been obtained (e.g., [2,3,[6][7][8][9][10]13,19,20,24], and the references cited therein).
Let us consider a function ϕ : [a, b] → [a, b] where [a, b] ⊂ R. Youness have defined the ϕ-convex functions in [22]: and t ∈ [0, 1], the following inequality holds: Obviously, if the function ϕ is the identity, then the classical convexity is obtained from the previous definition. Many properties of the ϕ-convex functions can be found, for instance, in [4,5,15,16,22].
Recall also that a function f : I → R is called strongly convex with modulus c > 0, if for all x, y ∈ I and t ∈ (0, 1). Strongly convex functions have been introduced by Polyak in [14] and they play an important role in optimization theory and mathematical economics. Various properties and applications of them can be found in the literature (see [1,11,12,14], and the references cited therein).
In this paper, we introduce the notion of strongly ϕ h -convex functions defined in normed spaces and present some properties of them. In particular, we obtain a representation of strongly ϕ h -convex functions in inner product spaces and, using the methods of [1,12,15], we give a characterization of inner product spaces, among normed spaces, which involves the notion of strongly ϕ h -convex function. Finally, a version of Hermite-Hadamard-type inequalities for strongly ϕ h -convex functions is presented. This result generalizes the Hermite-Hadamard-type inequalities obtained by Sarikaya in [15] for strongly ϕ-convex functions, and for c = 0, coincides with the Hermite-Hadamard inequalities for ϕ h -convex functions proved by Sarikaya in [16].

Main result
In what follows, (X, . ) denotes a real normed space, D stands for a convex subset of X, ϕ : D → D is a given function and c is a positive constant. Let h : (0, 1) → (0, ∞) be a given function. We say that a function f : for all x, y ∈ D and t ∈ (0, 1). In particular, if f satisfies (3) with h(t) = t, h(t) = t s (s ∈ (0, 1)), h(t) = 1 t , and h(t) = 1, then f is said to be strongly ϕ-convex, strongly ϕ s -convex, strongly ϕ-Gudunova-Levin function and strongly ϕ-P-function, respectively. The notion of ϕ h -convex function corresponds to the case c = 0. We start with the following lemma which gives some relationships between strongly ϕ h -convex functions and ϕ h -convex functions in the case where X is a real inner product space (that is, the norm . is induced by an inner product: . :=< x|x >).
i.e., f : Proof Since f is strongly ϕ h 2 -convex on I, thus for x, y ∈ I and t ∈ (0, 1), we have Proof By definition of strongly ϕ h -convexity, the proof is obvious.

Lemma 2.4
Let (X, . ) be a real inner product space, D be a convex subset of X and c be a positive constant and ϕ : Using properties of the inner product and assumption h(t) ≤ t, t ∈ (0, 1), we obtain which gives that g is a ϕ h -convex function.
(ii) Since g is a ϕ h -convex function, and using the assumption h(t) ≤ t, t ∈ (0, 1), we get which shows that f is strongly ϕ-convex with modulus c. (iii) In a similar way, we can prove it. This completes the proof.
The following example shows that the assumption that X is an inner product space is essential in the above lemma.
Example. Let X = R 2 and h(t) = t, t ∈ (0, 1). Let us consider a function ϕ : R 2 → R 2 , defined by ϕ(x) = x for every x ∈ R 2 and x = max {|x 1 |, |x 2 |} for x = (x 1 , x 2 ). Take f = . 2 . Then g = f − . 2 is ϕ h -convex being the zero function. However, f is not strongly ϕ h -convex with modulus 1. Indeed, for x = (1, 0) and y = (0, 1), we have The assumption that X is an inner product space in Lemma 2.4 is essential. Moreover, it appears that the fact that for every ϕ h -convex function g : X → R the function f = g + c . 2 is strongly ϕ h -convex characterizes inner product spaces among normed spaces. Similar characterizations of inner product spaces by strongly convex, strongly h-convex and strongly ϕ-convex functions are presented in [1,12,15], respectively. Proof The implication (i) ⇒ (ii) follows by Lemma 2.4. To see that (ii) ⇒ (iii) take g = 0. Clearly, g is ϕ h -convex function, whence f = c . 2 is strongly ϕ h -convex with modulus c. Consequently, . 2 is strongly ϕ h -convex with modulus 1. Finally, to prove iii) ⇒ i) observe that by the strongly ϕ h -convexity of . 2 and assumption h 1 2 = 1 2 , we obtain

Theorem 2.5 Let (X, . ) be a real normed space, D be a convex subset of X and
for all x, y ∈ X. Now, putting u = ϕ(x) + ϕ(y) and v = ϕ(x) − ϕ(y) in (4), we have for all u, v ∈ X . Conditions (4) and (5) mean that the norm . 2 satisfies the parallelogram law, which implies, by the classical Jordan-Von Neumann theorem, that (X, . ) is an inner product space. This completes the proof.
Proof From the strong ϕ h -convexity of f , we have Integrating the above inequality over the interval (0, 1), we obtain In the first integral, we substitute x = tϕ(a) + (1 − t)ϕ(b). Meanwhile, in the second integral, we also use the substitution To prove the second inequality, we start from the strong ϕ h -convexity of f meaning that for every t ∈ (0, 1), one has Integrating the above inequality over the interval (0, 1), we get The previous substitution in the first side of this inequality leads to which gives the second inequality of (6). This completes the proof.
Proof Since f is strongly ϕ h -convex with respect to c > 0, we have that for all t ∈ (0, 1) and Multiplying both sides of (8) by (9), it follows that Integrating the inequality (10) with respect to t over (0, 1), we obtain If we change the variable x := tϕ(a) + (1 − t)ϕ(b), t ∈ (0, 1), we get the required inequality in (7). This proves the theorem. .
Multiplying both sides of (12) by (13), it follows that Integrating the above inequality over the interval (0, 1), we get  In the first integral, we substitute x = tϕ(a) + (1 − t)ϕ(b) and simple integrals calculated, we obtain the required inequality in (11).
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