Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

Abstract

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="script">S</a:mi> </a:math> denotes the class of functions <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:mi>f</d:mi> <d:mfenced open="(" close=")"> <d:mrow> <d:mi>z</d:mi> </d:mrow> </d:mfenced> <d:mo>=</d:mo> <d:mi>z</d:mi> <d:mo>+</d:mo> <d:msubsup> <d:mrow> <d:mo>∑</d:mo> </d:mrow> <d:mrow> <d:mi>n</d:mi> <d:mo>=</d:mo> <d:mn>2</d:mn> </d:mrow> <d:mrow> <d:mo>∞</d:mo> </d:mrow> </d:msubsup> <d:msub> <d:mrow> <d:mi>a</d:mi> </d:mrow> <d:mrow> <d:mi>n</d:mi> </d:mrow> </d:msub> <d:msup> <d:mrow> <d:mi>z</d:mi> </d:mrow> <d:mrow> <d:mi>n</d:mi> </d:mrow> </d:msup> </d:math> analytic and univalent in the open unit disk <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"> <h:mi mathvariant="double-struck">U</h:mi> </h:math> , then the logarithmic coefficients <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M4"> <k:msub> <k:mrow> <k:mi>γ</k:mi> </k:mrow> <k:mrow> <k:mi>n</k:mi> </k:mrow> </k:msub> <k:mfenced open="(" close=")"> <k:mrow> <k:mi>f</k:mi> </k:mrow> </k:mfenced> </k:math> of the function <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M5"> <o:mi>f</o:mi> <o:mo>∈</o:mo> <o:mi mathvariant="script">S</o:mi> </o:math> are defined by <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M6"> <r:mi mathvariant="normal">log</r:mi> <r:mfenced open="(" close=")"> <r:mrow> <r:mi>f</r:mi> <r:mfenced open="(" close=")"> <r:mrow> <r:mi>z</r:mi> </r:mrow> </r:mfenced> <r:mo>/</r:mo> <r:mi>z</r:mi> </r:mrow> </r:mfenced> <r:mo>=</r:mo> <r:mn>2</r:mn> <r:msubsup> <r:mrow> <r:mo>∑</r:mo> </r:mrow> <r:mrow> <r:mi>n</r:mi> <r:mo>=</r:mo> <r:mn>1</r:mn> </r:mrow> <r:mrow> <r:mo>∞</r:mo> </r:mrow> </r:msubsup> <r:msub> <r:mrow> <r:mi>γ</r:mi> </r:mrow> <r:mrow> <r:mi>n</r:mi> </r:mrow> </r:msub> <r:mfenced open="(" close=")"> <r:mrow> <r:mi>f</r:mi> </r:mrow> </r:mfenced> <r:msup> <r:mrow> <r:mi>z</r:mi> </r:mrow> <r:mrow> <r:mi>n</r:mi> </r:mrow> </r:msup> </r:math> . In the current paper, the bounds for the logarithmic coefficients <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M7"> <ab:msub> <ab:mrow> <ab:mi>γ</ab:mi> </ab:mrow> <ab:mrow> <ab:mi>n</ab:mi> </ab:mrow> </ab:msub> </ab:math> for some well-known classes like <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M8"> <cb:mi mathvariant="script">C</cb:mi> <cb:mfenced open="(" close=")"> <cb:mrow> <cb:mn>1</cb:mn> <cb:mo>+</cb:mo> <cb:mi>α</cb:mi> <cb:mi>z</cb:mi> </cb:mrow> </cb:mfenced> </cb:math> for <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M9"> <hb:mi>α</hb:mi> <hb:mo>∈</hb:mo> <hb:mfenced close="]" open="("> <hb:mrow> <hb:mn>0</hb:mn> <hb:mo>,</hb:mo> <hb:mn>1</hb:mn> </hb:mrow> </hb:mfenced> </hb:math> and <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M10"> <lb:mi mathvariant="script">C</lb:mi> <lb:msub> <lb:mrow> <lb:mi mathvariant="script">V</lb:mi> </lb:mrow> <lb:mrow> <lb:mtext>hpl</lb:mtext> </lb:mrow> </lb:msub> <lb:mfenced open="(" close=")"> <lb:mrow> <lb:mn>1</lb:mn> <lb:mo>/</lb:mo> <lb:mn>2</lb:mn> </lb:mrow> </lb:mfenced> </lb:math> were estimated. Further, conjectures for the logarithmic coefficients <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M11"> <rb:msub> <rb:mrow> <rb:mi>γ</rb:mi> </rb:mrow> <rb:mrow> <rb:mi>n</rb:mi> </rb:mrow> </rb:msub> </rb:math> for functions <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M12"> <tb:mi>f</tb:mi> </tb:math> belonging to these classes are stated. For example, it is forecasted that if the function <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M13"> <vb:mi>f</vb:mi> <vb:mo>∈</vb:mo> <vb:mi mathvariant="script">C</vb:mi> <vb:mfenced open="(" close=")"> <vb:mrow> <vb:mn>1</vb:mn> <vb:mo>+</vb:mo> <vb:mi>α</vb:mi> <vb:mi>z</vb:mi> </vb:mrow> </vb:mfenced> </vb:math> , then the logarithmic coefficients of <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML"

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