On the coefficients of some subclasses of univalent functions (Q5943063)

The paper deals with a very special subclass of univalent functions. Namely, for given \(\alpha\in\langle 0,1\rangle\) , \(\beta\in(0,1\rangle\) and \(\gamma\in\langle 0,1)\) let \(L(\alpha,\beta,\gamma)\) denote the class of holomorphic functions \(f(z)=z+\sum_{n=2}^\infty a_nz^n\) in the unit disc \(|z|<1\) which satisfy the condition: NEWLINE\[NEWLINE \left|\frac{f'(z)-1}{\alpha f'(z)+(1-\gamma)}\right|<\beta,\quad |z|<1. NEWLINE\]NEWLINE Further, let \(L_c^*(\alpha,\beta,\gamma)\) denote the subclass of \(L(\alpha,\beta,\gamma)\) of functions which have the form NEWLINE\[NEWLINEf(z)=z-c\frac{\beta(\alpha+1-\gamma)}{2(1+\alpha\beta)}z^2-\sum_{n=3}^\infty a_nz^n\qquad (a_n\geq 0)\tag{*} NEWLINE\]NEWLINE where \(c\in\langle 0,1\rangle\). NEWLINENEWLINENEWLINEA few simple properties and sharp estimates of the class \(L_c^*(\alpha,\beta,\gamma)\) are given: coefficient estimates, distortion theorems etc.; e.g. a function \(f\) given by (*) is in \(L_c^*(\alpha,\beta,\gamma)\) if and only if NEWLINE\[NEWLINE \sum_{n=3}^\infty n(1+\alpha\beta)a_n\leq(1-c)\beta(\alpha+1-\gamma) NEWLINE\]NEWLINE and the result is sharp. The methods of proofs are of elementary character.