×

Starlike, convex and close-to-convex functions of complex order. (English) Zbl 0941.30008

Let \(\lambda\) and \(b\) be fixed numbers such that \(\lambda\in [0,1]\), \(b\in\mathbb{C}\), \(\text{Re} b>0\) and let \(U=\{z:|z|<1\}\). The authors consider two classes \(P(\lambda,b)\) and \(R(\lambda,b)\) of functions \(f\) of the form \(f(z)= z-\sum^\infty_{n=2} a_nz^n\), \(a_n\geq 0\) for \(n=2,3,\dots\), holomorphic in \(U\) and satisfying the condition \[ \text{Re} \left\{1+ {1\over b} \left({zf'(z) +\lambda z^2f''(z)\over (1-\lambda) f(z)+\lambda xf'(z)}-1\right) \right\}>0 \] or \[ \text{Re} \left\{1+ {1\over b}\bigl(f'(z) +\lambda zf''(z)-1 \bigr) \right\}>0, \] respectively. They obtain a few rather simple theorems relating to coefficient estimates for functions of the classes and for radii of close-to-convexity, starlikeness and convexity. Further, applications of the fractional derivative of order \(\delta\) for functions of these classes are given.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite