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Some sufficient conditions for univalence and close-to-convexity

Nunokawa Mamoru, Cho Nak Eun, Kwon Oh Sang, Sokół Janusz

Journal of Applied Analysis

2020

Vol. 26, nr 1

91--95

In the class of analytic functions in the unit disc |z| < 1 we prove some new sufficient conditions for functions to be univalent or to be close-to-convex in the unit disc. Also we extend Ozaki’s condition that Re{exp(iα)f (p)(z)} > 0 in |z| < 1 implies that f(z) is at most p-valent in |z| < 1.

Length problems for Bazilevič functions

Nunokawa Mamoru, Sokół Janusz, Tang Huo

Demonstratio Mathematica

2019

Vol. 52, nr 1

56--60

Let C(r) denote the curve which is image of the circle |z|=r<1 under the mapping f. Let L(r) be the length of C(r) and A(r) the area enclosed by the curve C(r). Furthermore M(r) = max|z|=r |f(z)|. We present some relations between these notions for Bazilevič functions.