New subclass of analytic functions defined by q -analogue of p -valent Noor integral operator

: In this paper, we introduce a certain subclass of analytic functions associated with q analogue of p -valent Noor integral operator in the open unit disc. A variety of useful properties for this subclass are investigated including coe ffi cient estimates and the familiar Fekete-Szeg¨o type inequalities. Several known sequences of the main results are also highlighted.

(1. 6) We note that 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 [20]). The q-number shift factorial for any non-negative integer n is defined by [n] q ! = 1 f or n = 0 [1] q [2] q · · · [n] q f or n ∈ N.
The Pochhammer q-generalized symbol for x > 0 and n ∈ N is also [x, q] n = 1 f or n = 0 [x] q [x + 1] q · · · [x + n − 1] q f or n ∈ N, and for x > 0, the q-gamma function is defined by For λ > −p (p ∈ N), we define the function f −1 λ+p−1,q (z) by where the function f λ+p−1,q (z) is given by [λ + p, q] n−p [1, q] n−p z n . (1.8) It is clear that the function defined in (1.8) converges absolutely in U. Using the idea of convolution we define the q-p-valent Noor integral operator I λ+p−1 q : A(p) −→ A(p) as follows: Φ q (λ, p, n)a n z n , (1.9) where Φ q (λ, p, n) = [p + 1, q] n−p [λ + p, q] n−p (λ > −p, p ∈ N). We note that: (i) For p = 1, we have the q-Noor integral operator I λ q f (z) ( f ∈ A) which was introduced and studied by Arif et al. [4]; which is the p-valent Noor integral operator (see [11]); (iii) Taking p = 1 and letting q −→ 1 − in (1.9), we obtain Noor integral operator for univalent functions (see [13,14]); (iv) For λ = 1, we have I p q f (z) = f (z) and for λ = 0, we have n p a n z n = z f ′ (z) p .

Geometric interpretation
where or, equivalently, (2.1) The boundary ∂Ω k of the above set when k = 0 becomes the imaginary axis, when 0 < k < 1 a hyperbola, when k = 1 a parabola and an ellipse when 1 < k < ∞. The functions p k (z) are defined by is the Legendre's complete elliptic integral of the first kind, and R ′ (t) is complementary integral of R(t) (see [9,10,18]).
By giving a specific value to the parameters q, λ, p, k, and b in the class ST q (λ, p, k, b), we get a lot of new and known subclasses studied by various others, for example, Also we note that: [20] ); [15][16][17] We need the following lemmas in order to establish our main results.
Lemma 2.1. [8] Let 0 ≤ k < ∞ be fixed and let p k be defined by (2.2). If p k (z) = 1 + Q 1 z + Q 2 z 2 + · · · , then c n z n ∈ P, i.e., let h be analytic in U and satisfies Re (h(z)) > 0 The result is sharp for a function given by c n z n ∈ P, then where v < 0 or v > 1, the equality holds iff h(z) = 1+z 1−z or one of its rotations. If 0 < v < 1, then he equality holds iff h(z) = 1+z 2 1−z 2 or one of its rotations. If v = 0, then he equality holds iff or one of its rotations. If v = 1, then he equality holds if and only if g is reciprocal of one of the function such that the equality holds in the case of v = 0.
Also the above upper bound is sharp, and it can improved as follows when 0 < v < 1 : and
Proof. Assume the inequality (3.1) holds. Let us assume that We have The last inequality is bounded by 1 if (3.1) holds. □ The inequality (3.2) is sharp for the function Choosing p = 1 and b = 1 − α, 0 ≤ α < 1, in Theorem 3.1, we obtain the following corollary.
in Theorem 3.1, we obtain the following consequence. Corollary 3.4. Let f ∈ A(p) be given by (1.1) and satisfy Then f ∈ ST q (λ, p, k, α).
Letting q −→ 1 − in Theorem 3.1, we obtain the following corollary.
in Theorem 3.1, we obtain the following consequence.
Taking p = 1 in Theorem 3.1, we obtain the following corollary.
Corollary 3.9. If a function f ∈ A has the form (1.1) (with p = 1) and satisfy Then f ∈ ST q (λ, k, b).

4)
and for all n ≥ 3
Proof. Let where p(z) = 1 + ∞ n=1 c n z n is analytic in U and it can be written as Comparing the coefficients of z n+p−1 on both sides of (3.6), we obtain Taking the absolute value on both sides and using |c n | ≤ Q 1 (n ≥ 1) (see [18]), we obtain (3.7) We apply the mathematical induction on (3.7), so for n = 2, we have this shows that the result is true for n = 2. Now for n = 3 we have which is true for n = 3. Let us assume that (3.7) is true for n ≤ t, that is Consider So, the result is true for n = t + 1. Also, we proved that the result true for all n(n ≥ 2) using mathematical induction. □ Taking p = 1 in Theorem 3.10, we obtain the following corollary.
in Theorem 3.10, we obtain the following result. Corollary 3.13. Let f ∈ A(p) be given by (1.1). If f ∈ ST q (λ, p, k, α), then and for all n ≥ 3, in Theorem 3.10, we obtain the following consequence.
Corollary 3.14. Let f ∈ A(p) be given by (1.1). If f ∈ ST q (λ, p, k, α), then and for all n ≥ 3, Proof. Let w 0 ∈ C and w 0 0 such that f (z) w 0 for z ∈ U. Then Since f 1 is univalent, so Now using Theorem 3.10, we have and hence we have This completes the proof of Theorem 3.15 □ Theorem 3.16. Let 0 ≤ k < ∞ be fixed and let f ∈ ST q (λ, p, k, b) with the form (1.1). Then for a complex µ, we have where Q 1 and Q 2 are given by (2.3) and (2.4), respectively. The result is sharp.
□ Putting p = 1 in Theorem 3.16, we obtain the following consequence.
Corollary 3.17. Let 0 ≤ k < ∞ be fixed and let f ∈ ST q (λ, k, b) with the form (1.1) (with p = 1). Then for a complex parameter µ, we have where Q 1 and Q 2 are given by (2.3) and (2.4), respectively. The result is sharp.
in Theorem 3.16, we get the following corollary. Corollary 3.19. Let 0 ≤ k < ∞ be fixed and let f ∈ ST q (λ, p, k, α) with the form (1.1). Then for a complex parameter µ, we have where Q 1 and Q 2 are given by (2.3) and (2.4), respectively. The result is sharp.
in Theorem 3.16, we get the following corollary.
Corollary 3.20. Let 0 ≤ k < ∞ be fixed and let f ∈ ST γ q (λ, p, k, α). Then for a complex parameter µ, we have The result is sharp.
Theorem 3.21. Let If f given by (1.1) belong to the class The result is sharp.
Proof. Applying Lemma 2.3 to (3.12) and (3.13), we can obtain our results asserted by Theorem 3.21. □ Putting p = 1 in Theorem 3.21, we obtain the following corollary.

Conclusions
Studies of the coefficient problems including the Fekete-Szegö problems continue to motivate researchers in Geometric Function Theory of Complex Analysis. In our present investigation, we have introduced and studied a new class ST q (λ, p, k, b) of analytic functions associated with q-analogue of p-valent Noor integral operator in the open unit disc U. For functions in this class, we have derived the coefficient estimates of the coefficients a p+1 and a n+p+1 for n ≥ 3, and Fekete-Szegö functional problems for functions belonging to this new class. Several of new results are shown to follow upon specializing the parameters involved in our main results.