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- W2113172029 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) $$r_alpha = (1 + sqrt {2alpha } )^{ - {1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}}$$ , 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 $$left| {1 + alpha } right|left| {a_3 - mu a_2^2 } right| leqslant ({2 mathord{left/ {vphantom {2 3}} right. kern-nulldelimiterspace} 3})max left[ {1,left| {1 + 2alpha - {3 mathord{left/ {vphantom {3 {2mu }}} right. kern-nulldelimiterspace} {2mu }}(1 + alpha )} right|} right].$$ ." @default.
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- W2113172029 date "1975-12-01" @default.
- W2113172029 modified "2023-10-16" @default.
- W2113172029 title "On a linear combination of some expressions in the theory of the univalent functions" @default.
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- W2113172029 doi "https://doi.org/10.1007/bf01472573" @default.
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