Abstract
In this note, we consider a subclass H3/2(p) of starlike functions f with f″(0)=p for a prescribed p∈[0,2]. Usually, in the study of univalent functions, estimates on the Taylor coefficients, Fekete–Szegö functional or Hankel determinats are given. Another coefficient problem which has attracted considerable attention is to estimate the moduli of successive coefficients |an+1|−|an|. Recently, the related functional |an+1−an| for the initial successive coefficients has been investigated for several classes of univalent functions. We continue this study and for functions f(z)=z+∑n=2∞anzn∈H3/2(p), we investigate upper bounds of initial coefficients and the difference of moduli of successive coefficients |a3−a2| and |a4−a3|. Estimates of the functionals |a2a4−a32| and |a4−a2a3| are also derived. The obtained results expand the scope of the theoretical results related with the functional |an+1−an| for various subclasses of univalent functions.
Highlights
As usual, denote by A the family of all normalized analytic functions ∞ f (z) = z + ∑ an zn (1)n =2 defined in the open unit disk U = {z ∈ C : |z| < 1} and let S be the subset of univalent functions in A
Let P be the class of analytic functions p with a positive real part in U, satisfying the condition p(0) = 1
We begin this section by finding the absolute values of the first three initial coefficients in the function class H3/2 ( p)
Summary
In 1997, Silverman [2] investigated the properties of a subclass of A, defined in terms of the quotient In [2], Silverman proved that all functions in Gb are starlike of order 2/(1 + 1 + 8b). Estimates of the difference of moduli of successive coefficients, for certain subclasses of S∗ , were obtained by Z. The upper bounds of the same funtionals | a3 − a2 | and | a4 − a3 | for various subclasses of univalent functions were obtained by. Motivated by the results given in [11,12,13], in the present paper we obtain upper bounds of the initial coefficients and upper bounds of | a3 − a2 | and | a4 − a3 | for a refined subclass of H3/2 , defined by. The second functional is a particular case of the generalized Zalcman functional, investigated by Ma [18], Efraimidis and Vukotić [19] and many others (see [20,21,22,23])
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