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

DOI: 10.5556/j.tkjm.43.2012.445-453

Keywords: Differential operator , Analytic functions , superordination

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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 $U=left{{z,inmathbb{C}:,left| z ight|,<,1} ight}$ and satisfy egin{equation*} {mathop{ m Re} olimits} left( {mu _{lambda _1 ,lambda _2 }^{n,m} f(z)} ight)^prime > 0,,,,,,,,,,(z in U). end{equation*} This paper focuses on attaining sharp upper bound for the functional $left| {a_2 a_4 - a_3^2 } ight|$ for functions $f(z)=z+ sumlimits_{k = 2}^infty {a_k ,z^k }$ belonging to the class $mathcal{H}_{lambda _1 ,lambda _2 }^{n,m}$.

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