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In its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this paper, we utilize some known Steffensen-type identities, obtained by using Lidstone's interpolating polynomial, to prove new generalizations of Steffensen's inequality. We obtain these new generalizations by using the weighted Hermite-Hadamard inequality for (2 n + 2) - convex and (2 n + 3) - convex functions. Further, the newly obtained inequalities can be observed as an upper- and lower-bound for utilized Steffensen-type identities. [ABSTRACT FROM AUTHOR]

In this paper we deal with functions related to generalized convexity and refine Jensen type inequalities satisfied by such functions. Specifically, we extend inequalities satisfied by uniformly convex functions, strongly convex functions as well as superquadratic functions. [ABSTRACT FROM AUTHOR]

In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions. [ABSTRACT FROM AUTHOR]

By using superquadracity, the new inequalities in this paper generalize a result of Hardy–Littlewood–Pólya and refine Jensen's type inequalities. Also, inequalities related to rearrangements of sets are obtained. [ABSTRACT FROM AUTHOR]

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain Fejér type inequalities for higher order convex functions and the general weighted integral formula involving w-harmonic sequences of functions. Also, Fejér type inequalities are obtained for the general three, four and five point quadrature formulae. Further, Fejér type inequalities for the corrected three, corrected four and corrected five point quadrature formulae are considered. In special cases, Fejér type estimates for Simpson, Maclaurin, corrected Simpson and corrected Maclaurin quadrature rules are derived. [ABSTRACT FROM AUTHOR]

The cosine function is a classical tool for measuring angles in inner product spaces, and it has various extensions to normed linear spaces. In this paper, we investigate a cosine function for the convex angle formed by two nonzero elements of a complex normed linear space, in connection with recent results on the Birkhoff-James approximate orthogonality sets. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of the $$\mathbb {K}$$ -Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite-Hadamard inequalities to the case when the usual Riemann integral is replaced by the $$\mathbb {K}$$ -Riemann integral and the convexity notion is replaced by $$\mathbb {K}$$ -convexity. [ABSTRACT FROM AUTHOR]

We observe that every quadrature of numerical integration together with a strictly increasing and continuous function generates a mean and then we study this family of means. We characterize the functions which generate weighted arithmetic means in this way and we show how to obtain comparison type theorems for means generated by different quadratures. [ABSTRACT FROM AUTHOR]

In this paper, some integral inequalities for uniformly convex functions are studied by using unordered submajorization for cumulative functions. Strongly convex functions and superquadratic functions are considered, too. A Levin–Stečkin like theorem is obtained for such functions. As applications, some bounds for the Fejér functional are derived. A result on the Schur-convexity of averages of convex functions is extended to uniformly convex functions. Some specifications for symmetric functions are also given. A corollary for symmetric probability density functions is established. A Levin–Stečkin type inequality for generalized ψ -uniformly convex functions is provided. Some interpretations for Simpson distributions are presented. [ABSTRACT FROM AUTHOR]

In this paper, the class of weakly ψ -uniformly convex functions is introduced and studied. Such functions are defined on a convex subcone D of a linear space V endowed with the structure of a normal decomposition system (V , G , (·) ↓) . In particular, superquadratic functions and c-strongly convex functions are recovered as special cases. Some results in Abramovich et al. (J Math Anal Appl 327: 1444–1460, 2007), intended for multidimensional superquadratic functions defined on R + m , are extended to the case of weakly ψ -uniformly convex functions. The concept of ψ -almost superadditivity of functions is presented and applied. Interpretations of the obtained results are demonstrated for some matrix normal decomposition systems. [ABSTRACT FROM AUTHOR]

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if $$Sp\left( A \right) \subset \left( 1,\infty \right) $$ , then and if $$Sp\left( A \right) \subset \left( 0,1 \right) $$ , then for each positive operator A and $$x\in \mathcal {H}$$ with $$\left\| x \right\| =1$$ . [ABSTRACT FROM AUTHOR]

In this paper, some fractional integral inequalities involving m-convex functions are established. The presented results are generalizations of the obtained inequalities in Dragomir and Toader (Babeş-Bolyai Math 38:21-28, 1993). [ABSTRACT FROM AUTHOR]

The article presents a paper that establishes inequalities of Hammer-Bullen type for convex sequences and nondecreasing convex function. The inequalities were established through the use of Wu-Debnath's method. The inequality of Fejér-Hammer-Bullen type for convex sequences was derived by assuming symmetry of an involved sequence.

Stochastic convexity and its applications are very important in mathematics and probability (Aequationes Mathematicae 20:184-197, 1980). There are two well-known inequalities for convex stochastic processes: Jensen's inequality and Hermite-Hadamard's inequality. Recently, Hafiz (Stoch Anal Appl 22:507-523, 2004) has provided fractional calculus for some stochastic processes. The problem is how to formulate these inequalities for stochastic processes in the class of fractional calculus and that is what is done in this paper. Our results generalize the corresponding ones in the literature. [ABSTRACT FROM AUTHOR]

The main objective of this article is to introduce a new class of real valued functions that include the well-known class of $${m}$$ -convex functions introduced by Toader (). The members of this collection are called Jensen $${m}$$ -convex and are defined, for $${m \in (0,1]}$$ , via the functional inequality where $${c_{m} := \frac{m+1}{m}}$$ . These functions generate a new kind of functional convexity that is studied in terms of its behavior with respect to basic algebraic operations such as sums, products, compositions, etc. in this paper. In particular, it is proved that any starshaped Jensen convex function is Jensen $${m}$$ -convex. At the same time an interesting example (Example 3) shows how the classes of Jensen $${m}$$ -convex functions depend on $${m}$$ . All the techniques employed come from traditional basic calculus and most of the calculations have been done with Mathematica 8.0.0 and validated with Maple 15 as well as all the figures included. [ABSTRACT FROM AUTHOR]

The paper gives answer to two open questions related to Mulholland's inequality. First, it is shown that there exists a larger set of solutions to Mulholland's inequality compared to the one delimited by Mulholland's condition. Second, it is demonstrated that the set of functions solving Mulholland's inequality is not closed with respect to compositions. [ABSTRACT FROM AUTHOR]

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [ABSTRACT FROM AUTHOR]

In this work, Levinson type inequalities involving two types of data points are proved using Green functions and the Hermite interpolating polynomial for the class of n-convex functions. In seek of applications to information theory some estimates for new functionals are obtained, based on f -divergence. Moreover, some inequalities involving Shannon entropies are deduced as well. [ABSTRACT FROM AUTHOR]

Let I be an interval. We consider the non-monotonic convex self-mappings $$f:I\rightarrow I$$ such that $$f^2$$ is convex. They have the property that all iterates $$f^n$$ are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every $$n\ge 2$$ there exists a convex iterative root $$f^{1/2^n}$$ of order $$2^n$$ and the sequence $$\{f^{1/2^n}\}$$ satisfies the condition of compatibility, that is $$ f^{1/2^n}\circ f^{1/2^n}= f^{1/2^{n-1}}.$$ A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions $$\{f^t,t>0\}$$ such that $$f^t\circ f^s=f^{t+s}, \ \ t,s >0$$ and $$f^1=f$$ . We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions. [ABSTRACT FROM AUTHOR]

Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite-Hadamard type. Namely, we determine all numbers $${a,\alpha,\beta\in[0,1]}$$ such that for all convex functions f the inequality is satisfied and all $${a,b,c,\alpha\in(0,1)}$$ with a + b + c = 1 for which we have . [ABSTRACT FROM AUTHOR]