Estimates of analytical functions whose ranges of values are contained in a circular lune

Abstract

The article introduces a class of analytical functions in a unit circle that have missing terms in the power series expansion. Their range of values is contained in a circular lune located in the right half-plane relative to the imaginary axis and symmetric relative to the real axis, one of the vertices of which is located at point 0. In this class of functions, the problem of finding the exact upper boundary of the modulus of the logarithmic derivative and the boundaries from below and above the real part and the modulus of the function is solved. Such results have always served as the basis for solving a number of extreme problems on subclasses of functions fz), analytical in the unit circle and normalized by the condition f (0) = f '(0) -1 = 0 . Some particular cases, when a circular lune degenerates into a circle or angle, yield well-known estimates established by such authors as T.H. MacGregor, R.M. Goel, D.B. Shaffer, and G.M. Shah and used by many researchers for decades to solve extreme problems. As an example of applications of the main result, the radius of starlikeness of one wide class of doubly close-to-starlike functions is obtained, which in particular cases gives a number of well-known results obtained in recent years.

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