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For maps of analytic functions f : D → D; f(0) = a ∈ D, we prove their Lipschitz continuity regarding the distance ratio metric with a Lipschitz constant C depending only on a. Question of the best possible C = C(a) remains open.

We give some new inequalities concerning the differences of power means. .

We give another global upper bound for Jensen's discrete inequality which is better than already existing ones. For instance, we determine a new converses for generalized A-G and G-H inequalities.

We give a generalization of well-known Valiron-Titchmarsh theorem on entire functions with negative zeros. Namely, we prove that (-1)plogP(r)ЄER[p,p+1] where P(r) denotes the canonical product of an entire function with genus p and almost negative zeros and ER is the class of extended regular variation.

For the power series distribution, generated by an entire function of finite order, we obtain the asymptotic behavior of its regularly varying moments. Namely, we prove that EwX??(X)\sim(EwX)??(EwX), ? > 0 (w??), where ?(?) is an arbitrary slowly varying function.

We prove a theorem concerning asymptotic behavior of general complex-valued kernel convolutions with slowly varying functions in the sense of Karamata. In applications we showed that the content of some classical theorems can be naturally extended on some parts of complex z-plane.

The classes KR∞, MR∞, R∞ of rapidly varying functions are natural extensions of Karamata's concept of regular variation. In ref. 2 we introduced a new class K of perfect Karamata's kernels and its subclasses Θ and Σ. In this paper we study inclusion properties of these classes and, among other results, we prove KR∞ C MR∞ C Σ C Θ C K. .

We give a necessary and sufficient condition for a positive measurable kernel C(-) to satisfy r f(t)c(t)dt -/(x) r c(t)dt (x -oo) Ji Jj whenever Ji is from the class of Karamata's regularly varying functions.

We give a unified solution of problems posed by Karamata--Blanusa--Mitrinovic more than 50 years ago on the determination of algebraic functions with prescribed asymptotic and monotonicity properties. .