Certain subclasses of analytic functions associated with fractional \(q\)–calculus operators. (English) Zbl 1229.33027

The authors introduce the \(q\)-analogue of the Riemann-Liouville fractional derivative and integral operators. The \(q\)-analogue of the differentiation and integration of a class of power functions is computed. Using the Riemann-Liouville \(q\)-fractional operators, some new classes of analytic and univalent functions on the open disc in the complex plane are defined. For these function classes coefficient inequalities and distortion theorems are proved. These results generalize those known from the \(q\)-theory of analytic functions.


33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
26A33 Fractional derivatives and integrals
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