Coefficient bounds for some classes of starlike functions

Abstract

Let t be given, 1/4 ^ t ^ oo, and let 5(0 denote the class of normalized starlike univalent functions / in | z | < 1 satisfying (i) |/(z)/z|^ί, |z|< 1, if 1/4^*^ 1, (ii) |/(z)/z 1 ^ f, \z\< 1, if 1 < t ^ oo. If f(z) = 2 + ΣΓ= 2 flfcZ fc G S(ί) and n is a fixed positive integer, then the authors obtain sharp coefficient bounds for | a n \ when t is sufficiently large or sufficiently near 1/4. In particular a sharp bound is found for | a-κ | when 1/4 ^ t ^ 1 and 5 ^ ί ^ oo. Also a sharp bound for \a 4 \ is found when l/4^ί^l or 12.259 =i t ^oo.

The Sharp Bound for Some Coefficient Functional within the Class of Holomorphic Bounded Functions and Its Applications

“…In [1] Barnard and Lewis proved a very interesting result that, if f ∈ S * M , then , where a * 1 ∈ (0.12, 0.13), a * 2 ∈ (0.23, 0.24) are the unique roots of the equations 175a 3 − 305a 2 + 148a − 14 = 0 500a 3 − 1011a 2 + 609a − 94 = 0, respectively.…”

“…The corresponding sharp result is presented in Theorem 1. We also give the applications for the class of holomorphic bounded and nonvanishing functions in the unit disk as well as for the class of bounded starlike functions, where we apply some deep result of Barnard and Lewis [1].…”

“…Proof. The estimate (37) was given in [1] and follows from (35) and (36) by direct calculations. In order to obtain (38) we apply Theorem A and the result of Barnard and Lewis (34).…”