Pure-Power Extremals in Sakaguchi Classes with Even-Vanishing Subordination: b_3-Free Bounds and a Structural Conjecture

Abstract

In this paper, we study the class  } of Sakaguchi-type starlike functions for a general univalent  with . We first prove that the coefficient recurrence derived in [22] and used in subsequent works contains a structural error: for odd , the correct leading factor is  rather than , a distinction forced by the evenness of . The corrected recurrence holds for every .

Writing , we isolate the even-vanishing property  1) as the mechanism that decouples the Carathéodory parameters controlling  from those controlling  and . Under the hypotheses  and  satisfied by every  and, more generally, for every  with  we establish the sharp bounds

together with the Fekete-Szegó inequality . None of these bounds depends on ; each is attained by a pure-power Schwarz function .

For the exponential class , where  and the even-vanishing property fails, we show that the Fekete-Szegö threshold shifts from  to  and the parameter decoupling ceases to hold. We conjecture that the pure-power extremal phenomenon persists for all even-vanishing Sakaguchi classes.