Coefficient inequalities for strongly close-to-convex functions
Let \(f(z)=z+a_2 z^2+a_3 z^3+ \cdots\) be a normalized, strongly close-to-convex function of order \(\alpha\) on the unit disk. This means that there exist a normalized convex univalent function \(\varphi\) and a real number \(\beta\) such that \(|\arg {f'(z) \over e^{i\beta} \varphi'(z)} |<{\alpha\pi \over 2}\). The authors prove two sharp coefficient inequalities for functions \(f\) in this class, namely, \(|a_3-a^2_2 |+ {1\over 3} |a_2 |^2 \leq{1\over 3} (1+4\alpha+ 2\alpha^2)\) and \(|a_3- {2\over 3} a^2_2|+ {2\over 3} |a_2 |^2\leq {1\over 3} (3+4 \alpha+ 2\alpha^2)\). Equality occurs if and only if \(f\) is a rotation of the function \[ F_\alpha (z)= {1\over 2(1+ \alpha)} \left[\left( {1+z\over 1-z} \right)^{1+ \alpha}- 1\right]= z+(1+\alpha) z^2+ \textstyle {1\over 3} (3+4 \alpha+2 \alpha^2)z^3+ \cdots. \] The functionals \(L(f)= \text{Re}(a^2_2-a_3)+ {1\over 3} |a_2|^2\) and \(M(f)= \text{Re}(3a_3 -2a^2_2) +2|a_2 |^2\) play an important part in the proof. The paper concludes with a formulation of the inequalities for nonnormalized, strongly close-to-convex functions of order \(\alpha\).
scientific article; zbMATH DE number 1005674