Initial logarithmic coefficients for functions starlike with respect to symmetric points
Paweł Zaprawa
Abstract
Abstract In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> and $${\mathcal {K}}_S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> and $${\mathcal {K}}_S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> and $${\mathcal {K}}_S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> .