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Abstract. We prove certain new inequalities for special means of two arguments, including the identric, arithmetic, and geometric means.

The purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problemwhere 0 < α < 1, D α 0+ denotes the Riemann-Liouville fractional derivative of order α, and f : (0, ∞) × R → R is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder's fixed point theorem enables to provide sufficient conditions on f , ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.MSC 2010 : Primary 34A08; Secondary 26A33, 34A12

The paper deals with the polynomial-like iterative functional equationBy using Schauder's fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.Mathematics Subject Classification (2010). Primary 39B12; Secondary 26A18, 47H10.

Abstract. The classical Hermite-Hadamard inequality characterizes the continuous convex functions of one real variable. The aim of the present paper is to give an analogous characterization for functions of a vector variable. The Hermite-Hadamard inequalityIn a letter sent on November 22, 1881, to the journal Mathesis (and published there two years later), Ch. Hermite [10] noted that every convex function f :The left-hand side inequality was rediscovered ten years later by J. Hadamard [7]. Nowadays, the double inequality (1) is called the Hermite-Hadamard inequality. The interested reader can find its complete story in the historical note by D. S. Mitrinović and I. B. Lacković [12]. The Hermite-Hadamard inequality has evoked the interest of many mathematicians. Especially in the last three decades it has been intensively investigated and generalized in several directions. For instance, a dual Hermite-Hadamard inequality was discussed by C. P. Niculescu [13], while a complete extension of (1) to the class of n -convex functions was obtained by M. Bessenyei and Zs. Páles [1]. Likewise, M. Bessenyei and Zs. Páles [2] proved a generalization of (1) for real-valued functions defined on an open interval I ⊆ R , which are convex with respect to a so-called positive regular pair over I . For an account on various results dealing with the Hermite-Hadamard inequality, the reader is referred to the monograph by S. S. Dragomir and C. E. M. Pearce [5].In what follows, we are concerned with Hermite-Hadamard-type inequalities for functions of a vector variable. As pointed out by C. P. Niculescu [15], the Choquet theory (see [17]) provides the framework for a natural extension of (1) to such functions. For the reader's convenience, we briefly present here this extension. Let E be a real locally

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Abstract. In this paper we prove several Ky Fan type inequalities involving certain StolarskyTobey means. (2000): 26D15. Mathematics subject classification