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Recently, Tremblay et al. invented a very interesting transformation formula for fractional derivatives of arbitrary order. Also, the authors have obtained a new generalized Leibniz rule and a corresponding integral analogue for the fractional derivatives of the product of two functions. In this paper, we apply the new transformation formula on the classical generalized Leibniz rule and the corresponding integral analogue due to Osler and on those established by the authors. Some special cases are given. [ABSTRACT FROM AUTHOR]

In the present paper, sharp upper bounds of | a3 - μa²2 | for the functions f (z) = z + a2z² + a3z³ + ... belonging to a new subclass of Sakaguchi type functions are obtained. Also, application of our results for subclass of functions defined by convolution with a normalized analytic function are given. In particular, Fekete-Szegõ inequalities for certain classes of functions defined through fractional derivatives are obtained. [ABSTRACT FROM AUTHOR]

In this paper, some oscillation criteria for solutions of a general second order non-linear differential equations with damping of the form (a(t) ψ (x (t)) k (x' (t)))' + p (t) k (x' (t)) + q (t) f (x (t)) = 0, are given. The results obtained extend some existing results in the literature by using the refined integral averaging technique introduced by Rogovchenko and Tuncay). [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of meromorphic functions sharing a small function with its derivative and prove one theorem. The the-orem improves the results of Jin-Dong Li and Guang-Xin Huang [10]. [ABSTRACT FROM AUTHOR]

Let ' be an analytic self-map of open unit disk D and Ψ is analytic on D. Then a weighted composition operator induced by ' with weight Ψ is given by (WΨ,φ'f) (z) = Ψ(z)f('(z)) for z in D and f analytic on D. For each p = 1, let Sp be the space of analytic functions on D whose derivatives belong to the Hardy space Hp. Given WΨ,φ : Aαp(D) ? Sq(D), using Carleson measure we characterize boundedness and compactness of WΨ,φ for 1 = p, q = ∞. [ABSTRACT FROM AUTHOR]