Convolution Properties for Certain Classes of Analytic Functions Defined byq-Derivative Operator

Abstract

We investigate convolution properties and coefficients estimates for two classes of analytic functions involving the -derivative operator defined in the open unit disc. Some of our results improve previously known results.

Convolution properties for a family of analytic functions involvingq-analogue of Ruscheweyh differential operator

“…Varying the parameters µ, b, A and B in the last Theorem, we get the following known results discussed earlier in [21]. Then the function f ∈ K q [A, B].…”

“…Taking µ = 0 and b = γ (q + 1) with γ ∈ C \ {0} in H b q (µ, A, B), we obtain the class S * q (γ, A, B) and it is a special form of the one studied in [20]. Further for γ = 1, we have the class [21] and the family S * q (1, 1, −1) ∼ = S * q was studied in [11]. Also we see that…”

On Fekete-Szegö Problems for Certain Subclasses Defined byq-Derivative

“…Quite a number of great mathematicians studied the concepts of -derivative, for example, by Gasper and Rahman [3], Aral et al [4], Li et al [5], and many others (see [6][7][8][9][10][11][12][13][14][15]). …”

SUBCLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDER ASSOCIATED WITH q􀀀MITTAG LEFFLER FUNCTION

“…for a function f which is differentiable in a given subset of C. Further, for p = 1, we have D q,1 f (z) = D q f (z) (see Seoudy and Aouf [12]). Making use of the q-derivative operator D q,p (0 < q < 1, p ∈ N) given by (1.2), we introduce the subclass S * q (p, α) of p-valently q-starlike functions of order α in U and the subclass C q (p, α) of p-valently q-convex functions of order α in U, 0 α < 1, as follows:…”