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In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a stability result for symmetrized convex stochastic processes are proved. [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 this paper, we establish a new class of dynamic inequalities of Hardy's type which generalize and improve some recent results given in the literature. More precisely, we prove some new Hardy-type inequalities involving many functions on time scales. Some new discrete inequalities are deduced in seeking applications. [ABSTRACT FROM AUTHOR]

In this paper, the author discusses a generalization of Schröder's and Abel's functional equation of the form ∑ n α n (x) f (u n (x)) = g (x). In fact, using the theory of operators and infinite matrices, we show under certain conditions this equation has only a unique bounded solution f : X → R. [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]

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]

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013-2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow-Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conditions on nonnegative rd-continuous weight functions u x and υ x which ensure that a reverse dynamic inequality of the form ∫ a ∞ f p (x) υ x Δ x 1 p ≤ C ∫ a ∞ u x ∫ a σ x K σ x , σ y f (y) Δ y q Δ x 1 q , holds when q ≤ p < 0 and 0 < q ≤ p < 1. Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig. [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]

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 order to provide an alternative proof to an inequality of Hilbert, Hardy proved This inequality and its relatives triggered research activity concerning norm inequalities including new inequalities by Copson and Levinson. In this paper, three theorems are established from which related inequalities of Hardy, Copson, and Levinson and various extensions are deduced as corollaries. [ABSTRACT FROM AUTHOR]

We use properties of Gegenbauer polynomials to prove that the formula ∑ k = 0 n λ k λ n - k (x + y) 2 k (x - y) 2 n - 2 k = 2 λ n ∑ k = 0 n n k n - λ - 1 / 2 k k - λ - 1 / 2 k x 2 k y 2 n - 2 k (0.1) is valid for all integers n ≥ 0 and complex numbers λ , x , y . The special case λ = - 1 / 2 was proved in 2014 by Gavrea and Ivan. We apply (0.1) and related results to deduce further identities. Among others, we present combinatorial identities involving harmonic numbers and trigonometric functions, and we prove an inequality for a certain cosine sum. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f-divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder's and Minkowski's inequalities. [ABSTRACT FROM AUTHOR]

A classical result states that for two continuous, strict means M , N : I 2 → I (I is an interval) there exists a unique (M, N)-invariant mean K : I 2 → I , i.e. such a mean that K ∘ (M , N) = K and, moreover, the sequence of iterates ((M , N) n) n = 1 ∞ converge to (K, K) pointwise. Recently it was proved that continuity assumption cannot be omitted in general. We show that if K is a unique (M, N)-invariant mean then, without continuity assumption, (M , N) n → (K , K) . [ABSTRACT FROM AUTHOR]

We present various new inequalities for cosine and sine sums. Among others, we prove that 0.1 0 ≤ ∑ k = 0 n (a) 2 k (2 k) ! cos ((2 k + 1) x) 2 k + 1 (a ∈ R) is valid for all n ≥ 0 and x ∈ [ 0 , π / 2 ] if and only if a ∈ [ - 2 , 1 ] , and that 0.2 0 ≤ ∑ k = 0 n (b) 2 k (2 k) ! sin ((2 k + 1) x) 2 k + 1 (b ∈ R) holds for all n ≥ 0 and x ∈ [ 0 , π ] if and only if b ∈ [ - 3 , 2 ] . Here, (a) n = ∏ j = 0 n - 1 (a + j) denotes the Pochhammer symbol. Inequality (0.1) with a = 1 is due to Gasper. We use it to obtain an integral inequality in the complex domain and to provide a one-parameter class of absolutely monotonic functions. An application of (0.2) leads to a new extension of the classical Fejér–Jackson inequality. [ABSTRACT FROM AUTHOR]

Walter and Weckesser's result (Aequationes Math 46:212-219, 1993), extending the Bushell-Okrasiński convolution type inequality (Bushell and Okrasiński in J Lond Math Soc (2) 41:503-510, 1990), gave some general conditions on the functions k:0,d→R and g:0,∞→R under which, for every increasing function f :  0,d→0,∞, the inequality ∫0xkx-sgfsds≤g∫0xfsds,x∈0,d,is satisfied. Applying the result on a simultaneous system of functional inequalities, we prove that if d>1, then, in general, both k and g must be power functions. [ABSTRACT FROM AUTHOR]

Several applications of Abel's partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated with a strong order unit, it is shown that the convergence of the series $$\sum x_{n}$$ implies the convergence in density of the sequence $$(nx_{n})_{n}$$ to 0. This is done by extending the Koopman-von Neumann characterization of convergence in density. Also included is a new proof of the Jensen-Steffensen inequality based on Abel's partial summation formula and a trace analogue of the Tomić-Weyl inequality of submajorization. [ABSTRACT FROM AUTHOR]