On a linear combination of some expressions in the theory of the univalent functions

Abstract

LetH(α) denote the class of regular functionsf(z) normalized so thatf(0)=0 andf′(0)=1 and satisfying in the unit discE the condition

$$\operatorname{Re} \left\{ {(1 - \alpha )f'(z) + \alpha (1 + zf''(z)/f'(z))} \right\} > 0$$

for fixed α. It is known thatH(0) is a particular class NW of close-to-convex univalent functions. The authors show the following results:Theorem 1. Letf(z)∈H(α). Thenf(z)∈NW if α≤0 andz∈E.Theorem 2. Letf(z)∈NW. Thenf(z)∈H(α) in |z|=r<r _α where i)$r_\alpha = (1 + \sqrt {2\alpha } )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$, α≥0 and ii)$r_\alpha = \sqrt {\frac{{1 - \alpha - \sqrt {\alpha (\alpha - 1)} }}{{1 - \alpha }}}$, α<0. All results are sharp.Theorem 3. Iff(z)=z+a ₂ z ²+a ₃ z ³+... is inH(α) and if μ is an arbitrary complex number, then

$$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$

.