The Pre-Schwarzian Norm Estimate for Analytic Concave Functions
The Pre-Schwarzian Norm Estimate for Analytic Concave Functions
Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi mathvariant="double-struck">D</mml:mi></mml:mrow></mml:math>denote the open unit disk and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow></mml:math>denote the class of normalized univalent functions which are analytic in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi mathvariant="double-struck">D</mml:mi></mml:mrow></mml:math>. Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mtext>C</mml:mtext><mml:mtext>o</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>be the class of concave functions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="script">S</mml:mi></mml:math>, which have the condition that the opening angle of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>f</mml:mi><mml:mo …