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The class of harmonic convex functions has acquired a very useful and significant placement among the non-convex functions, since this class not only reinforces some major results of the class of convex functions, but also has supported the development of some remarkable results in analysis where the class of convex functions is silent. Therefore, many researchers have deployed themselves to explore valuable results for this class of non-convex functions. This paper obtains new discrete inequalities for univariate harmonic convex functions on linear spaces related to a Jensen-type and a variant of the Jensen-type results. Our results are refinements of very important recent inequalities presented by Dragomir and Baloch et al. Furthermore, we provide the natural applications of our results. [ABSTRACT FROM AUTHOR]

We shall discuss three generalized moduli such as generalized modulus of convexity, modulus of smoothness, and modulus of Zou-Cui of quasi-Banach spaces and give some important properties of these moduli. Furthermore, we establish relationships of these generalized moduli with each other. [ABSTRACT FROM AUTHOR]

The aim of this report is to study modulus of convexity δB of a quasi-Banach space B. We prove that δB is convex, continuous, nondecreasing and for arbitrary uniformly convex quasi-Banach space B, δB ... We also prove that a quasi-Banach space B is uniformly convex if and only if δB(∈) ≥ 0. Moreover we prove that a non-trivial quasi-Banach space B is uniformly non-square if and only if δB(∈) > 0. [ABSTRACT FROM AUTHOR]

The aim of this report is to study modulus of convexity δB of a quasi-Banach space B. We prove that δB is convex, continuous, nondecreasing and for arbitrary uniformly convex quasi-Banach space B, δB(∊) = 1 - 1/C√1 - ∊2C2/4 . We also prove that a quasi-Banach space B is uniformly convex if and only if δB(∊) ≥ 0. Moreover we prove that a non-trivial quasi-Banach space B is uniformly non-square if and only if δB(∊) > 0. [ABSTRACT FROM AUTHOR]