The radius of univalence is found for the convolution <mml:math alttext="$f * g$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>&#8727;</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:math> of functions <mml:math alttext="$f \in S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:math> (normalized univalent functions) and <mml:math alttext="$g \in C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math> (close-to-convex functions). A lower bound for the radius of univalence is also determined when <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> and <mml:math alttext="$g$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi></mml:math> range over all of <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math>. Finally, a characterization of <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> provides an inclusion relationship.

International Journal of Mathematics and Mathematical Sciences

Univalence for convolutions

Abstract

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