Fractional Calculus for Certain Integral Operator Involving Logarithmic Coefficients

Mohammed, Aabed; Darus, Maslina; Breaz, Daniel

doi:10.3844/jms2.2009.118.122

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2012

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“…Note that this problem has yet to be solved for certain classes introduced in various studies (see for examples [5], [21], [22] and [23]). Note that Hankel problems have also been solved successfully for fractional operator which can be seen in [20].…”

Section: Resultsmentioning

confidence: 99%

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

Al-Abbadi

Darus

2012

Tamkang J. Math.

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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 $\mathcal{H}_{\lambda _1 ,\lambda _2 }^{n,m}$.

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Section: Resultsmentioning

confidence: 99%