Coefficient bounds for quotients of starlike functions
Let S denote the family of functions of the form \(f(z)=z+\sum^{\infty}_{2}a_ nz^ n\) analytic and univalent in the unit disk \(\Delta\). By K and \(S^*(\alpha)\) denote the subclasses of S whose elements are convex and starlike of order \(\alpha\), \(0\leq \alpha<1,\) respectively. Let \[ F(z)=wf(z)/(w- f(z))=z+\sum^{\infty}_{2}c_ nz^ n,\quad f\in S,\quad w\not\in f(\Delta). \] \textit{R. R. Hall} [Bull. Lond. Math. Soc. 12, 25-28 (1980; Zbl 0434.30012)] proved the boundedness of the coefficients of F(z) if \(f\in K\) by first showing that \(| z^ 2f'(z)f^ 2(z)|>4/\pi^ 2.\) The author proves the boundedness of the coefficients of F(z) if f belongs to any subfamily of S by merely showing that \(| z^ 2f'(z)f^ 2(z)| \geq B>0.\) He uses this result to prove the boundedness of the coefficients of F(z) when f belons to \(S^*(\alpha,M)\), \(\alpha>0\), the class of bounded starlike functions of order \(\alpha\), \(\alpha>0.\) The author [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] shows that functions of the form \(z+\sum^{\infty}_{2}a_ nz^ n\) are in \(S^*(\alpha)\) if \(\sum^{\infty}_{n=1}(n-\alpha)| a_ n| \leq 1-\alpha.\) For such type of class the author finds a sharp upper bound for the second coefficient as follows: Theorem. If the coefficients of \(f(z)=z+\sum^{\infty}_{2}a_ nz^ n\) satisfy the inequality \(\sum^{\infty}_{n=2}(n-\alpha)| a_ n| \leq 1- \alpha,\quad 0\leq \alpha \leq 1,\) then for any \(w\not\in f(\Delta)\) the function \(F(z)=wf(z)/(w-f(z))=z+\sum^{\infty}_{2}c_ nz^ n\) will satisfy \(| c_ 2| \leq(3-\alpha)/2.\) This result is sharp, with equality for \(f(z)=z-(1-\alpha)z^ 3/(3-\alpha)\) and \(w=2/(3- \alpha).\quad -\) When \(\alpha =0\), the bound in the above theorem is also attained for \(f(z)=z-z^ 2/2\) and \(w={1\over2}\). Note that for \(f(z)=z- z^ 2/2\) and \(w={1\over2}\), \((*)\quad F(z)=z+\sum^{\infty}_{2}((n+1)/2)z^ n.\) Consequently the author conjectures that the coefficients of F(z) as given in (*) are extremal for all functions \(f(z)=z+\sum^{\infty}_{2}a_ nz^ n\) which satisfy \(\sum^{\infty}_{2}n| a_ n| \leq 1.\) Finally the author offers the following: Conjecture. If \(f\in S\), then there exists a \(w\not\in f(\Delta)\) such that the coefficients for \(F(z)=wf(z)/(w- f(z))=z+\sum^{\infty}_{2}c_ nz^ n\) satisfy \(| c_ n| \geq 1.\) Equality holds for \(f(z)=z\) and \(w=1\).
class of bounded starlike functions of order \(\alpha\)
scientific article; zbMATH DE number 3865649