Gamma-Bazilevič Functions - Sa’adatul Fitri (Mathematics, 2020)
For <semantics> γ ≥ 0 </semantics> and <semantics> α ≥ 0 </semantics> , we introduce the class <semantics> <msubsup> B 1 γ </msubsup> ( α ) </semantics> of Gamma−Bazilevič functions defined for <semantics> z ∈ D </semantics> by <semantics> R e <mfenced separators="" open="{" close="}"> <mfenced separators="" open="[" close="]"> <mstyle displaystyle="true" scriptlevel="0"> z f ′ ( z ) f ( z ) 1 − α z α </mstyle> + <mstyle displaystyle="true" scriptlevel="0"> z f ″ ( z ) f ′ ( z ) </mstyle> + ( α − 1 ) <mfenced separators="" open="(" close=")"> <mstyle displaystyle="true" scriptlevel="0"> z f ′ ( z ) f ( z ) </mstyle> − 1 </mfenced> </mfenced> γ <mfenced open="[" close="]"> <mstyle displaystyle="true" scriptlevel="0"> z f ′ ( z ) f ( z ) 1 − α z α </mstyle> </mfenced> 1 − γ </mfenced> > 0 </semantics> . We shown that <semantics> <msubsup> B 1 γ </msubsup> ( α ) </semantics> is a subset of <semantics> B 1 ( α ) </semantics> , the class of <semantics> B 1 ( α ) </semantics> Bazilevič functions, and is therefore univalent in <semantics> D </semantics> . Various coefficient problems for functions in <semantics> <msubsup> B 1 γ </msubsup> ( α ) </semantics> are also given.