On a Generalized Convolution Operator

Recently in the paper [Mediterr. J. Math. 2016, 13, 1535–1553], the authors introduced and studied a new operator which was defined as a convolution of the three popular linear operators, namely the Sǎlǎgean operator, the Ruscheweyh operator and a fractional derivative operator. In the present paper, we consider an operator which is a convolution operator of only two linear operators (with lesser restricted parameters) that yield various well-known operators, defined by a symmetric way, including the one studied in the above-mentioned paper. Several results on the subordination of analytic functions to this operator (defined below) are investigated. Some of the results presented are shown to involve the familiar Appell function and Hurwitz–Lerch Zeta function. Special cases and interesting consequences being in symmetry of our main results are also mentioned.

We denote a subclass of H[0, 1] by A whose members are of the form: Additionally, let K denote a subclass of A whose members are convex (univalent) in U which is equivalent to Further, let S * denote a subclass of A of starlike functions, which is symmetric to K by relation f ∈ K ⇔ z f ∈ S * .
For two functions p, q ∈ H(U), we say p is subordinate to q, or q is superordinate to p in U and write p(z) ≺ q(z), z ∈ U, if there exists a Schwarz function ω, analytic in U with ω(0) = 0, and |ω(z)| < 1, z ∈ U such that p(z) = q(ω(z)), z ∈ U. Furthermore, if the function q is univalent in U, then we have following symmetry: p(z) ≺ q(z) ⇔ p(0) = q(0) and p(U) ⊂ q(U).
For the purpose of this paper, we consider a generalized form of operator F which is defined by δ and φ(a, c; z) are, respectively, given by (8) and (11). For the function f of the form (1), we have which is in symmetry to the Carlson-Schaffer operator L(a, c) given in (10), (11). By putting a = 2, c = 2 − λ and δ = 0 in (12), we have the series representation which also arises by the application of the differintegral operator Ω λ z defined in [8], (see also [3]) to the function f ∈ A. The operator in [3] does not yield the Srivastava-Attiya operator (9), and therefore the operator defined above by (12) is not symmetric to the operator defined in [3].
For m ∈ Z and δ > 0, the operator F m δ (a, c) in (12) is the one that was used in [1,9] in slightly varied forms. Furthermore, the operator F m δ (a, c) generalizes various known operators used in Geometric Function Theory and we exhibit such relationships here : Cȃtaş [12], Srivastava and Attiya [5], F 0 δ (a, c) = L(a, c), Carlson and Schaffer [6], Jung et al. [13].
For appropriate values of the parameters in (12) when a = 1 + ν (ν > −1), b = 2 − λ (λ < 2), δ = 1 and m = n + 1 (n ∈ N 0 ), we can obtain the operator defined in [14,15]. It is interesting to note that one requires the convolution of only two known linear operators as defined in (12) to define the various operators discussed above including a symmetric operator introduced recently in [14] which is a convolution of three well-known operators. Our aim in this paper is to study some subordination properties of the generalized operator F = F m δ (a, c). Several results are established using well-known lemmas and some of the results also involve the Appell function and the Hurwitz-Lerch Zeta function. Special cases and interesting consequences of our main results are also mentioned.
The class P [A, B] was introduced and studied by Janowski [16] and, in particular, we denote

Key Lemmas
To obtain our results, we need the following lemmas.
The function q is convex univalent and is the best (a, n)-dominant in the sense that if p(z) ≺ q 1 (z), then q(z) ≺ q 1 (z).
The Gaussian hypergeometric function 2 Following results for the function 2 F 1 (a, b; c; z) are well-known.

Lemma 3 ([21]
). Let F, G ∈ H(U) be any convex univalent functions in U. If f ≺ F and g ≺ G, is convex univalent.
Lemma 6 is a special case of Theorem 2.12 or Theorem 2.13 contained in [7]. Making use of Lemma 4, we proved the following result.

Main Results
We begin by finding a subordination property of the operator F which is contained in the following theorem.
Proof. Let p(z) = (F f (z)) , then p ∈ H [1,1] and with the use of the identity: which is a subordination by a convex univalent function. Applying Lemma 1, changing suitably the variables and making use of the identities (i) and (ii) of Lemma 2, we obtain that and the function q(z) is the best dominant. Using now the series expansion: for each of the binomial factors occurring in the last integrand of (22) and changing the order of summation and integration (permissible under the assumed conditions mentioned above) and interpreting the resulting series in terms of the Appell function defined above by (20), we get p(z) ≺ q(z) = F 1 [1/δ; −l, l; 1/δ + 1; −Cz, −Dz].
This proves the result that where q(z) is a convex univalent function.
On using the relation (13), we obtain the following result from Theorem 1 provided that (a) > 0.
Proof. Let P(z) = (F f (z)) , then P ∈ H [1,1] and with the use of the identity: we have where Q is a convex univalent function.

Corollary 2.
Let a function f ∈ A and −1 ≤ F < E ≤ 1. If (a) > 0, the operator F m δ (a + 1, c) satisfies the condition (24), then Furthermore, where The result is best possible.
Additionally, on applying a special case when a = 1, we obtain from the expression (27), the following result involving the Sȃlȃgean operator D m for m ∈ N 0 with the use of the identity: 2 F 1 1, 1; 2;
We next prove the following theorem using Lemma 5.

Theorem 3.
Let a function f ∈ A and −1 ≤ F < E ≤ 1. If (a) > 0 and the operator F m δ (a + 1, c) satisfies the condition (24), then where 0 ≤ α < 1. Using the identity (26), we write which yields the result Hence, on putting t = uz (z ∈ U) and on using Lemma 5 with the condition (31), we obtain that where 0 ≤ γ < 1. This proves Theorem 3 on putting the value of α from (31).
Before stating and proving our next result, we recall the Hurwitz-Lerch Zeta function Φ(z; s; a) defined by for some s ∈ C \ Z − 0 , a > 0, [26]. We consider Φ(z, s, a) also for |z| = 1 when s > 1.
Proof. Observe that if n = 1, the result is trivial. For n ≥ 2 and for i = 1, 2, . . . , n, let Then ω i are analytic in U with ω i (0) = 1. Using (33), we obtain Hence, by Lemma 1 (for the case when n = γ = 1), we have where q i is convex in U and is the best dominant. Applying now Lemma 3 to the subordination (36) for i = 1, 2, . . . , n − 1, and to the subordination (33) for i = n, we obtain where h is convex in U being the convolution of functions which are convex in U and is given by (35) in terms of the Hurwitz-Lerch Zeta function (32). The left-hand side of the above subordination in (37) is evidentially This proves Theorem 4.
Results on the convolution of finite number of analytic functions have also been investigated earlier by considering a different operator in [27] (see also [4,28,29]).

Theorem 5.
Let a function f ∈ A and assume that |A|, |B|, |C|, |D| ≤ 1. If and Proof. We first observe that we have subordination under a convex univalent function in both (39) and in (40). Therefore applying Lemma 3, we obtain In view of (12), the left-hand side turns out to be the operator {F f (z)} , while the right-hand side is where q is convex in U and is the best dominant. Now on applying Lemma 3 to the subordination conditions (45) and (39), we obtain The left-hand side simplifies to {F f (z)} , while the right-hand side is where θ is given by (47). This proves the result (46).
Lastly, we prove the following result.

Theorem 7.
Let a function f ∈ A and assume that ν(z) is convex univalent and 0 < c ≤ a. If a ≥ 2 or a + c ≥ 3 and operator F defined in (12) satisfies