Duality for convolution on subclasses of analytic functions and weighted integral operators

: In this article, we investigate a class of analytic functions de ﬁ ned on the unit open disc z z : 1 (cid:2) { ∣ ∣ } = < , such that for every f β γ , α (cid:3) ( ) ∈ , α 0 > ,

Abstract: In this article, we investigate a class of analytic functions defined on the unit open disc z z holds.We find conditions on the numbers α β , , and γ such that , where SP λ ( ) denotes the set of all λ-spirallike functions.We also make use of Ruscheweyh's duality theory to derive conditions on the numbers α β γ , , and the real-valued function φ so that the integral operator V f φ ( ) ) into the set SP λ ( ), provided φ is non-negative normalized function ( ) ∈ − (see, e.g., [1][2][3][4][5][6]).Every analytic function f in is a convex λ-spirallike function on if and only if zf z ( ) ′ is a λ-spirallike function on .For α β 0, 0 1 > ≤ ≤ , and γ 0 1 < ≤ , we denote by β γ , α ( ) the set of all functions f in provided For f and g in (or 0 , g g z f : ; , the convolution product (Hadamard product) of f and g, denoted by f g * , is a function in (or 0 ) defined by The second dual or the dual hull of is defined by ( ) = * * * * .Indeed, we have, ⊆ * * (see, e.g., [7] and [8]).
Then, the Gaussian hypergeometric function is defined by .
The classical theory of integral transforms and their applications have been studied for a long time, and they are applied in many fields of mathematics [28][29][30][31][32]. Several integral transforms are extended to various spaces of distributions [29], tempered distributions [33], distributions of compact support [34], ultradistributions [35], and many others.In a classical sense, Fourier and Ruscheweyh introduced the integral transform V : where φ is a real-valued integrable function satisfying the normalizing condition [4] In the literature, the integral operator V f φ ( ) has been discussed by many authors on various choices of φ (see, e.g., [9,11,12,36,37]).In what follows, we introduce the function g g α γ λ ,

≔
as a solution to the differential equation tg t e te e t e e t t However, we find conditions on α β , , and γ so that | | < .We refer to the monographs [38] and [10] for more details on a variety of sufficient conditions on the λ-spirallike functions.
In Section 2, we present several lemmas which simplify our results.Section 3 is devoted for our main results and applications.One more result establishes the inclusion In another conclusion, we impose conditions on the set P β γ , α ( ) to be univalent.Several remarks, corollaries, and theorems are also derived in some detail.

Preliminary lemmas
The following are preliminary lemmas which are very useful in our next analysis.
Then, for all continuous linear functionals φ on we have φ φ x e e 1 2 Then, we have ,R e 1 2 2 1 0, , 1 ≠ , and V β γ , be given by (3).Then, we have , where Γ , Γ 1 2 are continuous linear functionals on V β γ , with V Γ 0 which is the requirement of the Duality Principle.Therefore, the Duality Principle can be stated with a slightly weaker condition, but more complicated, when V β γ , satisfies [7].In the present article, we apply the duality principle on the set V β γ , and hence we will not state it in its most general form.

Main results
In this section, we discuss various results involving spirallike functions, hypergeometric functions, and certain class of integral transforms.We immense our section by establishing the following theorem. where Proof.Let f be a function in the class β γ , α ( ) Hence, we have Therefore, we obtain a one-to-one correspondence between P β γ , α ( ) and V β γ , * * .Thus, by aid of Theorem 2.2, For z ∈ , let us consider the continuous linear functional λ : * * .Therefore, (6) holds if and only if Using the properties of the convolution, we reformulate (7) as For all z ∈ , x 1 | | = , the equality on the right side of (8) takes its value on the line w Re ) can be expressed in terms of Gaussian hypergeometric function as . , 1 . .
The following remark expresses a new form of Inequality (5).
Proof.Following the previous analysis, we write This indeed satisfies the above inequality when Therefore, we obtain Duality for convolution on subclasses of analytic functions and weighted integral operators  5 Theorem 3.4.
Then, the function f belongs to P 0, 1 α ( ) and, hence, it is univalent for By using change of variables, we rewrite ψ z ( ) in the form In view of these representations, we can write Thus, in view of (11), the preceding observation reveals Hence, equation ( 12) is equivalent to where , then G z Re 12 ( ) ≥ ∕ .Also, it is well-known that functions with real parts greater than 1/2 preserve the closed convex hull under convolution [10, p.23].Therefore, from (13) we write hold.Then, for every real λ, λ x e e 1 2 Hence, by applying equation (12), equation ( 15) is equivalent to Using a series expansion, we obtain ) Therefore, we can write ))

∞
and λ be a real number.Then, by λ SP( ) we denote the subclass of of all λ-spirallike functions for which λ ,

−
By substituting in the left-hand side of the previous inequality and using the change of variables υ st = in the resulting equation, it follows that This implies that our result is sharp for the λ-spirallike function.□4 Concluding remarksIn this article, a class of analytic functions was discussed on a unit open disc z conditions on the numbers α β , , and γ were imposed so that β γ , α () defines a subset of the set λ SP( ) of λ-spirallike functions for all λ , s duality theory was employed in predicting conditions on the numbers α β γ , , and the real-valued functions φ so that the integral transform V f φ ( ) maps β γ , α () into λ SP( ) for nonnegative and normalized functions φ.
be expressed in terms of an integral equation as By taking into account definitions and following simple computations we obtain ) again, integrating by part suggests the following compact form: