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Subclasses of \(k\)-uniformly convex and starlike functions defined by generalized derivative. I. (English) Zbl 1152.30309

Denote by \(\mathcal{S}\) the class of functions of the form \(f(z)=z+a_2z^2+\dots,\) analytic and univalent in the open unit disc \(U={z\colon |z|<1}\). Let \(D^n \colon \mathcal{S}\rightarrow\mathcal{S}\) be the differential operator defined by the reviewer [in Complex analysis – Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lecture Notes Math. 1013, 362–372 (1983; Zbl 0531.30009 )], that is, \(D^0f=f, D^1f(z)=Df(z)=zf'(z), D^nf=D(D^{n-1}f). \) Let \(f\in\mathcal{S},\) \(k\in[0,\infty]\) and \(n\in{0, 1, 2, \dots};\) we say that the function \(f\) belongs to the class \(\mathcal{T}(k,n)\) iff \[ \operatorname{Re}\left(\frac{D^{n+1}f(z)}{D^n f(z)} \right )>k\left|\frac{D^{n+1}f(z)}{D^n f(z)} -1 \right |, z\in U. \] We remark that \(\mathcal{T}(k,1)\) is the class of \(k\)-uniformly convex functions and that \(\mathcal{T}(k,0)\) is the class of \(k\)-starlike functions defined and studied by S. Kanas and A. Wiśniowska [J. Comput. Appl. Math. 105, No. 1–2, 327–336 (1999; Zbl 0944.30008); Rev. Roum. Math. Pures Appl. 45, 647–657 (2000; Zbl 0990.30010)].
In the present paper some extremal problems for functions in \(\mathcal{T}(k,n)\) are solved.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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