Inclusion and properties neighbourhood for certain p-valent functions associated with complex order and q–p-valent Cătaş operator

Abstract

Here in our paper, we present two new subclasses of multivalent analytic functions and complex order by using q-p-valent Cătaş operator. Also we obtain coefficient estimates and consequent inclusion relationships involving the N p,q i,δ (f)-neighbourhood of these classes.

Properties of neighborhood for certain classes associated with complex order and m-q-p-valent functions with higher order

“…(1) lim q→1 − F n,m q,p (j, 1, σ , b, β) = S j (n, p, m, σ , b, β) see Aouf et al [22]; (2) F n,0 q,p (j, , σ , b, β) = S n q (j, , p, σ , b, β)see Aouf and madian [14], with ̺ = 0]; (3) lim q→1 − F 0,0 q,p (j, , σ , b, β) = S j (p, σ , b, β) see Aouf and Mostafa [19], with b k = 1; (4) lim q→1 − F n,m q,p (j, , σ , b, β) = F n,m p (j, , σ , b, β)…”

“…In specially, if we obtain Now, we define the q, j, δ, m −neighborhood for f ∈ T (p, j) by In particular, if h(z)given by (1.12), we immediately have We note that (i) N p,q j,δ,0 (f ) = N p,q j,δ (f )andN p,q j,δ,0 (h) = N p,q j,δ (h)(see [14,20]);…”

“…From (1.1) (with j = 1 ) and (1.3), we deduce that such that q-integer number k [k] q is defined by We observe that for a function fwhich is differentiable in a given subset of C . For all f (z) ∈ T (p, j), we find ( see [25] ) (1) D n ,q,p f (0) (z) = I n q,p ( )f (z),(Aouf and Madian [14], with [11,18], Cătaş [24], with l = 0 ] and [37]); (4) 26,27], with l = 0)); (5) [39] see also [10,12]); (7)…”

“…The generalization of derivative and integral in q-calculus is known as q-analogue derivative and q-analogue integral. Recently, many authors used the q-analogue derivative and q-analogue integral to generalize many classes and many operators in Geometric Function Theory (see, for example, [14,33,40,42]).…”

Certain Classes of Analytic Functions Associated With q-Analogue of p-Valent Cătaş Operator

“…for a function f which is differentiable in a given subset of C. For a function f (z) ∈ A j (p), Aouf and Madian [15] defined the q-analogue p-valent Cȃtas ¸operator…”