Remarks to properties of analytic in the unit disc functions P-valent along with their derivatives

Myroslav Sheremeta


Let K=(kj) be an increasing sequence of  integers, k0=0, and E(K) be a class of analytic in the disk D=z:z<1 functions f such that for some p all derivatives f(kj) are in the middle p-valent in D. It is proved that  every function fE(K) is entire if and only if


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W. K. Hayman, Multivalent functions, Cambridge University Press, 1958, 180 p.

S. M. Shah, Analytic functions with univalent derivatives and entire function of exponential type, Bull. Amer. Math. Soc. 78 (1972), no. 2, 110-118. DOI: 10.1090/S0002-9904-1972-12900-8

M. M. Sheremeta, Disproving of a Shah conjecture concerning univalent functions, Math. Stud, 2 (1993), 46-48 (in Ukrainian).



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