A study of sharp coefficient bounds for a new subfamily of starlike functions
Khalil Ullah, H. M. Srivastava, Ayesha Rafiq, Muhammad Arif, Sama Arjika
Abstract
Abstract In this article, by employing the hyperbolic tangent function tanh z , a subfamily $\mathcal{S}_{\tanh }^{\ast }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>S</mml:mi><mml:mo>tanh</mml:mo><mml:mo>∗</mml:mo></mml:msubsup></mml:math> of starlike functions in the open unit disk $\mathbb{D}\subset \mathbb{C}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>⊂</mml:mo><mml:mi>C</mml:mi></mml:math> : $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mi>z</mml:mi><mml:mo>:</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mtext> and </mml:mtext><mml:mo>|</mml:mo><mml:mi>z</mml:mi><mml:mo>|</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math> is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class $\mathcal{S}_{\tanh }^{\ast } $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>S</mml:mi><mml:mo>tanh</mml:mo><mml:mo>∗</mml:mo></mml:msubsup></mml:math> of starlike functions in $\mathbb{D}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> . In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.