On coefficient bounds of a certain class of p-valent \(\lambda\)-spiral functions of order \(\alpha\) (Q1089457)

In this paper the study of \textit{D. A. Patil} and the reviewer [Indian J. Pure Appl. Math. 10, 842-853 (1979; Zbl 0419.30013)] is extended. Let \(S^{\lambda}(A,B,p,\alpha)\), \((| \lambda | <\pi /2,-1\leq A<B\leq 1\), and \(0\leq \alpha <p)\) denote the class of functions \(f(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n\) holomorphic in the unit disc U of the complex plane that satisfy for \(z=re^{i\theta}\in U\), \[ (i)\quad e^{i\lambda}\sec \lambda (zf'/f)-ip \tan \lambda =(p+[pB+(A- B)(p-\alpha)]w(z))/(1+Bw(z)), \] (ii) w(z) is holomorphic in U with \(w(0)=0.\) (iii) and \(| w(z)| \leq | z|\) for \(z\in U\). Bounds on the coefficients \(a_ n\) are obtained. Further \(| a_{p+2}-\mu a^ 2_{p+1}|\) is maximized over the class \(S^{\lambda}(A,B,p,\alpha)\) for complex values \(\mu\).