Hankel determinant for certain class of analytic function defined by generalized derivative operator

Abstract

The authors in \cite{mam1} have recently introduced a new generalised derivatives operator $ \mu_{\lambda _1 ,\lambda _2 }^{n,m},$ which generalised many well-known operators studied earlier by many different authors. By making use of the generalised derivative operator $\mu_{\lambda_1 ,\lambda _2 }^{n,m}$, the authors derive the class of function denoted by $ \mathcal{H}_{\lambda _1 ,\lambda _2 }^{n,m}$, which contain normalised analytic univalent functions $f$ defined on the open unit disc $U=\left\{{z\,\in\mathbb{C}:\,\left| z \right|\,<\,1} \right\}$ and satisfy \begin{equation*}{\mathop{\rm Re}\nolimits} \left( {\mu _{\lambda _1 ,\lambda _2 }^{n,m} f(z)} \right)^\prime > 0,\,\,\,\,\,\,\,\,\,(z \in U).\end{equation*}This paper focuses on attaining sharp upper bound for the functional $\left| {a_2 a_4 - a_3^2 } \right|$ for functions $f(z)=z+ \sum\limits_{k = 2}^\infty {a_k \,z^k }$ belonging to the class $\mathcal{H}_{\lambda _1 ,\lambda _2 }^{n,m}$.

Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions

“…Motivated by the results obtained by different authors in this direction mentioned above, in the present paper we introduced a new generalized derivative operator M m α,β,λ f(z) and investigate the upper bound for functional |a 2 a 4 − a 2 3 | for the function f belonging to the class M m α,β,λ f(z). We first state some preliminary lemmas required for proving our results.…”

“…For example, Janteng et al [10] have studied the sharp bound for the function f in (1.1), consisting the functions which derivative has a positive real part and have the result |a 2 a 4 − a 2 3 | 4/9. The same author [11] obtained the result for the sharp upper bounds for starlike and convex functions as |a 2 a 4 − a 2 3 | 1 and |a 2 a 4 − a 2 3 | 1/8, respectively. The subclass M m α,β,λ f(z) is defined as the following.…”

“…Various authors studied and investigated the second Hankel determinant for a certain class of analytic functions such as Al-Refai and Darus [3], Abubaker and Darus [1], Al-Abbadi and Darus [2], and Bansal [5].…”