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

  • Ebrahim Amini , Mojtaba Fardi , Shrideh Al-Omari and Kamsing Nonlaopon EMAIL logo
Published/Copyright: January 30, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } , such that for every f ∈ P α ( β , γ ) , α > 0 , 0 ≤ β ≤ 1 , 0 < γ ≤ 1 , and ∣ z ∣ < 1 , the inequality

Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0

holds. We find conditions on the numbers α , β , and γ such that P α ( β , γ ) ⊆ S P ( λ ) , for λ ∈ ( − π 2 , π 2 ) , where S P ( 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 φ ( f ) maps the set P α ( β , γ ) into the set S P ( λ ) , provided φ is non-negative normalized function ( ∫ 0 1 φ ( t ) d t = 1 ) and

V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t .

Keywords: convolution; Hadamard product; duality principle; λ-spirallike functions; dual set; Gaussian hypergeometric function
MSC 2010: 05A15; 11B68; 26B10; 33E20

1 Introduction

Let A be the set of analytic functions defined on the open unit disc U = { z ∈ C ; ∣ z ∣ < 1 } possessing the property that f ( z ) = z + ∑ n = 2 ∞ a n z n and λ be a real number. Then, by SP ( λ ) we denote the subclass of A of all λ -spirallike functions for which λ ∈ − π 2 , π 2 (see, e.g., [1,2,3, 4,5,6]). Every analytic function f in A is a convex λ -spirallike function on U if and only if z f ′ ( z ) is a λ -spirallike function on U . For α > 0 , 0 ≤ β ≤ 1 , and 0 < γ ≤ 1 , we denote by P α ( β , γ ) the set of all functions f in A provided

Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 for all z ∈ U .

For f and g in A (or A 0 , A 0 = { g : g ( z ) = f ( z ) z ; f ∈ A } ), f ( z ) = ∑ n = 0 ∞ a n z n and g ( z ) = ∑ n = 0 ∞ b n z n , the convolution product (Hadamard product) of f and g , denoted by f ∗ g , is a function in A (or A 0 ) defined by

( f ∗ g ) ( z ) = ∑ n = 0 ∞ a n b n z n , z ∈ U .

For a set V ⊆ A 0 , the dual set V ∗ is defined as V ∗ = { g ∈ A 0 : ( f ∗ g ) ( z ) ≠ 0 , ∀ f ∈ V , z ∈ U } . The second dual or the dual hull of V is defined by V ∗ ∗ = ( V ∗ ) ∗ . Indeed, we have, V ⊆ V ∗ ∗ (see, e.g., [7] and [8]). Let ( x ) k denote the Pochhammer symbol [6]

( x ) 0 = 1 and ( x ) k = x ( x + 1 ) … ( x + k + 1 ) , for k ∈ N .

Then, the Gaussian hypergeometric function is defined by ([1–27])

F 1 2 ( a , b , c ; z ) = ∑ k = 0 ∞ ( a ) k ( b ) k ( c ) k k ! z k .

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 φ : A ⟶ A ,

V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t ,

where φ is a real-valued integrable function satisfying the normalizing condition [4]

∫ 0 1 φ ( t ) d t = 1 .

In the literature, the integral operator V φ ( f ) 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

g ( t ) + 1 − γ α γ t g ′ ( t ) = 2 − e − i λ − t e − i λ e − i λ ( 1 + t ) − 2 ( 1 − e − i λ ) e − i λ log ( 1 + t ) t ,

which can be expressed in terms of an integral equation as

(1) g ( t ) = α γ 1 − γ ∫ 0 1 s α γ 1 − γ − 1 2 − e − i λ − t e − i λ e − i λ ( 1 + t ) − 2 ( 1 − e − i λ ) e − i λ log ( 1 + t ) t d s .

However, we find conditions on α , β , and γ so that P α ( β , γ ) ⊆ SP ( λ ) , for given ∣ λ ∣ < π 2 . We refer to the monographs [38] and [10] for more details on a variety of sufficient conditions on the λ -spirallike functions. For detailed analysis of various integral operators we refer readers to [14,15,16, 17,18,19] and references cited therein. 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 P α ( β , γ ) ⊆ SP ( λ ) for α > 0 , 0 ≤ β < 1 , 0 < γ < 1 , λ being real number but ∣ λ ∣ < π 2 . In another conclusion, we impose conditions on the set P α ( β , γ ) to be univalent. Several remarks, corollaries, and theorems are also derived in some detail.

2 Preliminary lemmas

The following are preliminary lemmas which are very useful in our next analysis.

Lemma 2.1

(Duality principle, see [8]). Let V ⊆ A 0 be a compact subset having the property

(2) f ∈ V ⇒ ∀ ∣ x ∣ ≤ 1 : f x ∈ V , f x ( z ) = f ( x z ) .

Then, for all continuous linear functionals φ on A we have φ ( V ) = φ ( V ∗ ∗ ) and c o ¯ ( V ) ⊆ c o ¯ ( V ∗ ∗ ) , where c o ¯ stands for the closed convex hull of the set.

Lemma 2.2

[7] Let f ∈ A . Then, the function f belongs to SP ( λ ) if and only if

1 z ( f ( z ) ∗ h ( z ) ) ≠ 0 , z ∈ U ,

where

h ( z ) = z + x − e − 2 i λ 1 + e − 2 i λ ( 1 − z ) 2 , ∣ x ∣ = 1 .

Lemma 2.3

[13] Let 0 ≤ γ < 1 and β ∈ R such that β ≠ 1 . Let

(3) V β , γ = γ ( 1 − β ) 1 + x z 1 − x z + ( 1 − γ ) ( 1 − β ) 1 + y z 1 − y z + β , ∣ x ∣ = ∣ y ∣ = 1 , z ∈ U .

Then, we have

V β , γ ∗ = f ∈ A 0 : ∃ α ∈ R , Re g ( z ) − 1 − 2 β 2 ( 1 − β ) > 0 , g ( z ) = f x ( z ) , ∣ x ∣ ≤ 1

and

V β , γ ∗ ∗ = f ∈ A 0 ; Re g ( z ) − β 1 − β > 0 , g ( z ) = f x ( z ) , ∣ x ∣ ≤ 1 .

Lemma 2.4

Let 0 ≤ γ < 1 and β ∈ R , β ≠ 1 , and V β , γ be given by (3). Then, we have

Γ 1 ( g ) Γ 2 ( g ) = Γ 1 ( f ) Γ 2 ( f ) ,

for every g ∈ V β , γ ∗ and some f ∈ V β , γ , where Γ 1 , Γ 2 are continuous linear functionals on V β , γ with Γ 2 ( V β , γ ) ≠ 0 .

It is clear from the context that the set V β , γ in (3) does not satisfy the property (2), i.e., if f ∈ V β , γ then f ( x z ) ∈ V β , γ for all ∣ x ∣ ≤ 1 , 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.

3 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.

Theorem 3.1

Let α > 0 , 0 ≤ β < 1 , 0 < γ < 1 , ∣ x ∣ = 1 , and λ be a real number such that ∣ λ ∣ < π 2 . Then P α ( β , γ ) ⊆ SP ( λ ) if and only if

(4) F ( x , z ) = α γ ∑ k = 1 ∞ k ( x + 1 ) + 1 + e − 2 i λ ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k ,

where

(5) Re { F ( x , z ) } > 1 + cos 2 λ 2 ( 1 − β ) .

Proof

Let f be a function in the class P α ( β , γ ) and g α , γ ( z ) = f ′ ( z ) + 1 − γ α γ z f ″ ( z ) . Then, we have g α , γ ∈ V β , γ ∗ ∗ . If f ( z ) = ∑ k = 1 ∞ a k z k , a 1 = 1 , then

g α , γ ( z ) = 1 + ∑ k = 1 ∞ ( k + 1 ) a k + 1 + 1 − γ α γ k ( k + 1 ) a k + 1 z k = 1 + ∑ k = 1 ∞ ( k + 1 ) a k + 1 1 + 1 − γ α γ k z k = ∑ k = 0 ∞ ( k + 1 ) a k + 1 1 + 1 − γ α γ k z k = ∑ k = 1 ∞ k a k 1 + 1 − γ α γ ( k − 1 ) z k − 1 .

Hence, we have

f ( z ) z = ∑ k = 1 ∞ a k z k − 1 = g α , γ ( z ) ∗ ∑ k = 1 ∞ α γ k ( α γ + ( 1 − γ ) ( k − 1 ) ) z k − 1 .

Therefore, we obtain a one-to-one correspondence between P α ( β , γ ) and V β , γ ∗ ∗ . Thus, by aid of Theorem 2.2, P α ( β , γ ) ⊆ SP ( λ ) if and only if

(6) g α , γ ( z ) ∗ ∑ k = 1 ∞ α γ z k − 1 k ( α γ + ( 1 − γ ) ( k − 1 ) ) ∗ 1 + x − e − 2 i λ 1 + e − 2 i λ z ( 1 − z ) 2 ≠ 0 , ∀ g ∈ V , ∀ ∣ x ∣ = 1 , ∀ z ∈ U .

For z ∈ U , let us consider the continuous linear functional λ z : A 0 ⟶ C such that

λ z ( h ) = h ( z ) ∗ ∑ k = 1 ∞ α γ z k − 1 k ( α γ + ( 1 − γ ) ( k − 1 ) ) ∗ 1 + x − e − 2 i λ 1 + e − 2 i λ ( 1 − z ) 2 ≠ 0 .

By the Duality Principle, we obtain λ z ( V ) = λ z ( V β , γ ∗ ∗ ) . Therefore, (6) holds if and only if

1 + 2 ( 1 − β ) ∑ k = 1 ∞ z k ∗ 1 + ∑ k = 1 ∞ α γ z k ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k ∗ 1 + ∑ k = 1 ∞ k + 1 + x − e − 2 i λ 1 + e − 2 i λ k z k ≠ 0 .

Hence, we have

(7) 1 + 2 ( 1 − β ) α γ 1 + e − 2 i λ ∑ k = 1 ∞ k ( x + 1 ) + 1 + e − 2 i λ ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k ≠ 0 .

Using the properties of the convolution, we reformulate (7) as

(8) α γ ∑ k = 1 ∞ k ( x + 1 ) + 1 + e − 2 i λ ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k ≠ − 1 + e − 2 i λ 2 ( 1 − β ) .

For all z ∈ U , ∣ x ∣ = 1 , the equality on the right side of (8) takes its value on the line Re w ≠ − 1 + cos 2 λ 2 ( 1 − β ) . So (8) is equivalent to (4).□

Corollary 3.2

The function F ( x , z ) can be expressed in terms of Gaussian hypergeometric function as

F ( x , z ) = − ( 1 + e − 2 i λ ) + α γ ( e − 2 i λ − x ) α γ + γ − 1 1 z log 1 1 − z + − ( 1 − γ ) ( 1 + e − 2 i λ ) + α γ ( x + 1 ) α γ + γ − 1 F 1 2 1 , α γ 1 − γ , α γ 1 − γ + 1 ; z .

Proof

By taking into account definitions and following simple computations we obtain

F ( x , z ) = α γ 1 − γ ∑ k = 1 ∞ k ( x + 1 ) + 1 + e − 2 i λ ( k + 1 ) α γ 1 − γ + k z k = α γ ( e − 2 i λ − x ) α γ + γ − 1 ∑ k = 1 ∞ 1 k + 1 z k + − ( 1 − γ ) ( 1 + e − 2 i λ ) + α γ ( x + 1 ) α γ + γ − 1 ∑ k = 1 ∞ α γ 1 − γ α γ 1 − γ + k z k = − α γ ( e − 2 i λ − x ) α γ + γ − 1 + − ( 1 − γ ) ( 1 + e − 2 i λ ) + α γ ( x + 1 ) α γ + γ − 1 + α γ ( e − 2 i λ − x ) α γ + γ − 1 F 1 2 ( 1 , 1 , 2 ; z ) + − ( 1 − γ ) ( 1 + e − 2 i λ ) + α γ ( x + 1 ) α γ + γ − 1 F 1 2 ( 1 , α γ 1 − γ , α γ 1 − γ + 1 ; z ) .

That is,

F ( x , z ) = − α γ ( e − 2 i λ + 1 ) − ( 1 − γ ) ( 1 + e − 2 i λ ) α γ + γ + 1 + α γ ( e − 2 i λ − x ) α γ + γ − 1 i.e. = − ( 1 + e − 2 i λ ) + α γ ( e − 2 i λ − x ) α γ + γ − 1 1 z log 1 1 − z + − ( 1 − γ ) ( 1 + e − 2 i λ ) + α γ ( x + 1 ) α γ + γ − 1 F 1 2 ( 1 , α γ 1 − γ , α γ 1 − γ + 1 ; z ) . □

The following remark expresses a new form of Inequality (5).

Remark 3.3

The following inequality holds.

1 + cos 2 λ 2 α γ ( 1 − β ) + Re ∑ k = 1 ∞ 1 α γ + ( 1 − γ ) k z k ≥ ∑ k = 1 ∞ k ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k + ∑ k = 1 ∞ 1 ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k .

Proof

Following the previous analysis, we write

1 + cos 2 λ 2 α γ ( 1 − β ) + Re ∑ k = 1 ∞ k ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k + Re ∑ k = 1 ∞ 1 ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k ≥ Re − x ∑ k = 1 ∞ k ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k + Re − e − 2 i λ ∑ k = 1 ∞ 1 ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k .

This indeed satisfies the above inequality when

1 + cos 2 λ 2 α γ ( 1 − β ) + Re ∑ k = 1 ∞ 1 α γ + ( 1 − γ ) k z k ≥ ∑ k = 1 ∞ k ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k + ∑ k = 1 ∞ 1 ( k + 1 ) ( α γ + ( 1 − γ ) k ) z k .

Therefore, we obtain

1 + cos 2 λ 2 α γ ( 1 − β ) + Re ∑ k = 1 ∞ 1 α γ + ( 1 − γ ) k z k ≥ ∑ k = 1 ∞ 1 α γ + ( 1 − γ ) k z k . □

Theorem 3.4

Let f ∈ P α ( β , γ ) , α > 0 , 0 ≤ β ≤ 1 , and 0 < γ ≤ 1 . Define

(9) K α , γ = ∫ 0 1 d t 1 − t 1 − γ α γ .

Then, the function f belongs to P α ( 0 , 1 ) and, hence, it is univalent for

β ≥ 1 − 2 K α , γ 2 ( 1 − K α , γ ) .

Proof

Let α > 0 and γ > 0 . Define

ϕ ( z ) = 1 + ∑ n = 1 ∞ 1 + 1 − γ α γ n z n

and

ψ ( z ) = 1 + ∑ n = 1 ∞ α γ α γ + ( 1 − γ ) n z n = 1 + ∑ n = 1 ∞ z n 1 + 1 − γ α γ n = ∫ 0 1 1 1 − t 1 − γ α γ z d t .

By using change of variables, we rewrite ψ ( z ) in the form

(10) ψ ( z ) = 1 1 − z , γ = 1 , α γ 1 − γ ∫ 0 1 s α γ 1 − γ − 1 − 1 1 − s z d s , 0 < γ < 1 .

In view of these representations, we can write

(11) f ′ ( z ) + 1 − γ α γ z f ″ ( z ) = f ′ ( z ) ∗ ϕ ( z ) and f ′ ( z ) + 1 − γ α γ z f ″ ( z ) ∗ ψ ( z ) = f ′ ( z ) .

Now, we let f ∈ P α ( β , γ ) . Then, in view of the Duality Principle (see Lemma 2.1), we may restrict our attention to the function f ∈ P α ( β , γ ) for which

f ′ ( z ) + 1 − γ α γ z f ″ ( z ) = γ ( 1 − β ) 1 + x z 1 − x z + ( 1 − β ) ( 1 − γ ) 1 + y z 1 − y z + β .

Thus, in view of (11), the preceding observation reveals

(12) f ′ ( z ) = γ ( 1 − β ) 1 + x z 1 − x z + ( 1 − β ) ( 1 − γ ) 1 + y z 1 − y z + β ∗ ψ ( z ) .

Hence, equation (12) is equivalent to

(13) f ′ ( z ) = γ 1 + x z 1 − x z + ( 1 − γ ) 1 + y z 1 − y z ∗ ( ( 1 − β ) ψ ( z ) + β ) , = γ 1 + x z 1 − x z + ( 1 − γ ) 1 + y z 1 − y z ∗ ∫ 0 1 ( 1 − β ) 1 1 − t 1 − γ α γ z + β d t = γ 1 + x z 1 − x z + ( 1 − γ ) 1 + y z 1 − y z ∗ G ( z ) ,

where

G ( z ) = ∫ 0 1 ( 1 − β ) 1 1 − t 1 − γ α γ z + β d t .

Therefore, we have

Re G ( z ) ≥ ∫ 0 1 ( 1 − β ) 1 1 − t 1 − γ α γ + β d t = ( 1 − β ) K α , γ + β ,

where K α , γ satisfies (9). Note that if β ≥ ( 1 − 2 K α , γ ) ∕ 2 ( 1 − K α , γ ) , then Re G ( z ) ≥ 1 ∕ 2 . 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

f ′ ( z ) = γ 2 1 − x z − 1 ∗ G ( z ) + ( 1 − γ ) 2 1 − y z − 1 ∗ G ( z ) = 2 γ G ( x z ) − γ + 2 ( 1 − γ ) G ( y z ) − ( 1 − γ ) = 2 γ G ( x z ) + 2 ( 1 − γ ) G ( y z ) − 1 .

But, since R e f ′ ( z ) > 0 , we have f ∈ P α ( 0 , 1 ) .□

Theorem 3.5

Let f ∈ P α ( β , γ ) , α > 0 , 0 ≤ β ≤ 1 , 0 < γ ≤ 1 . Let the following integral equation

(14) β 1 − β = − ∫ 0 1 φ ( t ) g ( t ) d t

hold. Then, for every real λ , ∣ λ ∣ < π 2 , we have V φ ( f ) ∈ SP ( λ ) if and only if

Re ∫ 0 1 Π γ ( t ) h ( t z ) t z − − t + e − i λ ( 1 + t ) e − i λ ( 1 + t ) 2 d t ≥ 0 ,

where

Π γ ( t ) = ∫ t 1 Λ γ ( s ) s α γ 1 − γ − 2 d s , Λ γ ( t ) = ∫ t 1 φ ( σ ) α γ 1 − γ d σ ( γ > 0 )

and

h ( z ) = z + x − e − 2 i λ 1 + e − 2 i λ ( 1 − z ) 2 , ∣ x ∣ = 1 , z ∈ U .

Proof

Let α > 0 , γ > 0 , and F ( z ) = V φ ( f ) . Then F ∈ SP ( λ ) if and only if

(15) 0 ≠ F ( z ) z ∗ h ( z ) z = 1 z ∫ 0 1 φ ( t ) f ( t z ) t d t ∗ h ( z ) z = ∫ 0 1 φ ( t ) d t 1 − t z ∗ f ( z ) z ∗ h ( z ) z .

Hence, by applying equation (12), equation (15) is equivalent to

(16) ( 1 − β ) ∫ 0 1 φ ( t ) 1 z ∫ 0 z h ( t w ) t w − g ( t ) d w d t ∗ ψ ( z ) ∗ γ 1 + x z 1 − x z + ( 1 − γ ) 1 + y z 1 − y z ≠ 0 ,

where g ( t ) is defined by (1). Equation (16) indeed holds if and only if

Re ( 1 − β ) ∫ 0 1 φ ( t ) 1 z ∫ 0 z h ( t w ) t w − g ( t ) d w d t ∗ ψ ( z ) > 1 2 .

Now, by using the fact ( 1 − β ) ∫ 0 1 φ ( t ) ( 1 − g ( t ) ) d t = 1 we derive

Re ∫ 0 1 φ ( t ) 1 z ∫ 0 z h ( t w ) t w − 1 + g ( t ) 2 d w d t ∗ ψ ( z ) ≥ 0 .

Using the definition of ψ given by (10), the above expression becomes

Re ∫ 0 1 φ ( t ) 1 z ∫ 0 z ∫ 0 1 s α γ 1 − γ − 1 α γ 1 − γ h ( t w ) t w − 1 + g ( t ) 2 d s d w d t ≥ 0 .

By the definition of g presented in (1), it is easy to see that

1 + g ( t ) 2 = α γ 1 − γ ∫ 0 1 s α γ 1 − γ − 1 1 2 1 + 2 − e − i λ − e − i λ s t e − i λ ( 1 + s t ) − 2 ( 1 − e − i λ ) e − i λ log 1 + s t s t d s .

By substituting in the left-hand side of the previous inequality and using the change of variables υ = s t in the resulting equation, it follows that

(17) Re ∫ 0 1 φ ( t ) t α γ 1 − γ ∫ 0 t v α γ 1 − γ − 1 1 z ∫ 0 z h ( w v ) w v − t 2 1 + 2 − e − i λ − e − i λ v e − i λ ( 1 + v ) − 2 ( 1 − e − i λ ) e − i λ log 1 + v v d w d v d t ≥ 0 .

By integrating by parts, Inequality (17) yields

Re ∫ 0 1 Λ γ ( t ) t α γ 1 − γ − 2 1 z ∫ 0 z h ( t w ) w − t 2 1 + 2 − e − i λ − e − i λ t e − i λ ( 1 + t ) − 2 ( 1 − e − i λ ) e − i λ log 1 + t t d w d t ≥ 0 ,

where Λ γ ( t ) = ∫ t 1 φ ( σ ) σ α γ 1 − γ d σ . Once again, integrating by part suggests the following compact form:

Re ∫ 0 1 Π γ ( t ) h ( t z ) t z − − t + e − i λ ( 1 + t ) e − i λ ( 1 + t ) 2 d t ≥ 0 ,

where Π γ ( t ) = ∫ t 1 Λ γ ( s ) s α γ 1 − γ − 2 d s . Thus, F ∈ SP ( λ ) . To extend our result from the particular choice of f given in (12) to all of P α ( β , γ ) we refer to Lemma 2.4. Note that V φ defines a linear functional and the condition on the λ -spirallikeness Re e i λ z F ′ ( z ) F ( z ) > 0 > 0 is expressed as a quotient of two linear functionals. Therefore, Lemma 2.4 can be applied. Finally, to prove the sharpness, let f ∈ P α ( β , γ ) be in the form

f ′ ( z ) + 1 − γ α γ z f ″ ( z ) = γ ( 1 − β ) 1 + x z 1 − x z + ( 1 − γ ) ( 1 − β ) 1 + y z 1 − y z + β .

Let x = y = 1 . Using a series expansion, we obtain

f ( z ) = z + 2 ( 1 − β ) ∑ n = 2 ∞ α γ n ( α γ + ( n − 1 ) ( 1 − γ ) ) z n .

Therefore, we can write

F ( z ) = V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t = z + 2 ( 1 − β ) ∑ n = 2 ∞ α γ μ n n ( α γ + ( n − 1 ) ( 1 − γ ) ) z n ,

where μ n = ∫ 0 1 λ ( t ) t n − 1 d t . Furthermore, it is easy to write g ( t ) in (1) in a series expansion as

(18) g ( t ) = 1 + − 2 e − i λ ∑ n = 1 ∞ ( − 1 ) n n + 2 − e − i λ n + 1 α γ α γ + n ( 1 − γ ) t n .

Now by equations (14) and (18), we have

β 1 − β = − ∫ 0 1 φ ( t ) g ( t ) d t = − ∫ 0 1 φ ( t ) 1 + − 2 e − i λ ∑ n = 1 ∞ ( − 1 ) n n + 2 − e − i λ n + 1 α γ α γ + n ( 1 − γ ) t n d t = − 1 − − 2 e − i λ ∑ n = 1 ∞ ( − 1 ) n n + 2 − e − i λ n + 1 α γ μ n + 1 α γ + n ( 1 − γ ) .

Hence, we obtain

1 1 − β = − 2 e − i λ ∑ n = 2 ∞ ( − 1 ) n − 1 n + 1 − e − i λ α γ μ n n ( α γ + n ( 1 − γ ) ) .

Computing F ′ ( − 1 ) , indeed gives

F ′ ( − 1 ) = 1 + 2 ( 1 − β ) ∑ n = 1 ∞ ( − 1 ) n − 1 α γ μ n ( n − 1 ) ( 1 − γ ) + α γ = ( 1 − e − i λ ) + 2 ( 1 − β ) ( 1 − e − i λ ) ∑ n = 2 ∞ ( − 1 ) n − 1 α γ μ n n ( ( n − 1 ) ( 1 − γ ) + α γ ) = ( 1 − e − i λ ) 1 + 2 ( 1 − β ) ∑ n = 2 ∞ ( − 1 ) n − 1 α γ μ n n ( ( n − 1 ) ( 1 − γ ) + α γ ) = − ( 1 − e − i λ ) F ( − 1 ) .

Hence, we obtain

Re e i λ − F ′ ( − 1 ) F ( − 1 ) < 0 .

This implies that our result is sharp for the λ -spirallike function.□

4 Concluding remarks

In this article, a class of analytic functions was discussed on a unit open disc U = { z : ∣ z ∣ < 1 } . Certain conditions on the numbers α , β , and γ were imposed so that P α ( β , γ ) defines a subset of the set SP ( λ ) of λ -spirallike functions for all λ ∈ − π 2 , π 2 . Ruscheweyh’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 φ ( f ) maps P α ( β , γ ) into SP ( λ ) for nonnegative and normalized functions φ .

Acknowledgements

The authors would like to thank the handling editor and the referees for their helpful comments and suggestions.

  1. Funding information: This research has received funding support from the National Science, Research and Innovation Fund (NSRF), Thailand.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Jpn. Acad. Ser. A 69 (1993), 234–237. DOI: https://doi.org/10.3792/pjaa.69.234. 10.3792/pjaa.69.234Search in Google Scholar

[2] M. Nunokawa, O. S. Kwon, Y. J. Sim, and N. E. Cho, Sufficient conditions for Carathéodory functions, Filomat. 32 (2018), no. 3, 1097–1106. DOI: https://doi.org/10.1515/ms-2017-0290. 10.2298/FIL1803097NSearch in Google Scholar

[3] M. Nunokawa, and J. Sokol, New conditions for starlikeness and strongly starlikeness of order alpha, Houst. J. Math. 43 (2017), no. 2, 333–344. Search in Google Scholar

[4] M. Nunokawa, and J. Sokol, Some applications of first-order differential subordinations, Math. Slovaca 67, no. 4, 939–944, DOI: https://www.jstor.org/stable/24898719. 10.1515/ms-2017-0022Search in Google Scholar

[5] Q. Xu and S. Lu, The Alexander transformation of a subclass of spirallike functions of type β, J. Inequal. Pure Appl. Math. 10 (2009), no. 1, 1–17, DOI: https://doi.org/10.1007/s00025-018-0771-3. 10.1007/s00025-018-0771-3Search in Google Scholar

[6] M. Hussain, M. Raza, and J. Sokol, On a subclass of starlike functions, Results Math. 73 (2018), no. 22, 1–12, DOI: https://doi.org/10.1007/s00025-018-0771-3. 10.1007/s00025-018-0771-3Search in Google Scholar

[7] S. Ruscheweyh, Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc, Trans. Amer. Math Soc. 210 (1975), 63–74. DOI: https://doi.org/10.2307/1997122. 10.1090/S0002-9947-1975-0382626-7Search in Google Scholar

[8] S. Ruscheweyh, Convolutions in Geometric Function Theory, Sem. Math. Sup. vol. 83, Press Univ. Montreal, 1982. Search in Google Scholar

[9] Y. C. Kim and F. Ronning, Integral transforms of certain subclasses of analytic functions, Rocky Mountain J. Math. 24 (1994), no. 2, 529–538. DOI: https://doi.org/10.1006/jmaa.2000.7383. 10.1006/jmaa.2000.7383Search in Google Scholar

[10] S. S. Miller and P. T. Mocanu, Differential subordinations: theory and applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York, 2000. 10.1201/9781482289817Search in Google Scholar

[11] S. Ponnusamy and F. Ronning, Duality for Hadamard product applied to certain integral transforms, Complex Var. Theory Appl. 32 (1997), 263–287. DOI: https://doi.org/10.1080/17476939708814995. 10.1080/17476939708814995Search in Google Scholar

[12] S. Ponnusamy and F. Ronning, Integral transforms of a class of analytic functions, Complex var Theory Appl. 53 (2008), 423–434. DOI: https://doi.org/10.1080/17476930701685767. 10.1080/17476930701685767Search in Google Scholar

[13] H. Silverman, E. M. Silvia, and D. Telage, Convolutions conditions for convexity starlikeness and spiral-likeness, Math. Z. 162 (1978), no. 2, 125–130. DOI: https://doi.org/10.1007/BF01215069. 10.1007/BF01215069Search in Google Scholar

[14] S. Al-Omari, D. Baleanu, and D. Purohit, Some results for Laplace-type integral operator in quantum calculus, Adv. Differ. Equ. 124 (2018), 1–10. DOI: https://doi.org/10.1186/s13662-018-1567-1. 10.1186/s13662-018-1567-1Search in Google Scholar

[15] S. Al-Omari, On q-analogues of Mangontarum transform of some polynomials and certain class of H-functions, Nonlinear Studies 23 (2016), no. 1, 51–61. DOI: https://doi.org/10.1016/j.jksus.2015.04.008. 10.1016/j.jksus.2015.04.008Search in Google Scholar

[16] S. Al-Omari, On q-analogues of the Mangontarum transform for certain q-Bessel functions and some application, J. King Saud Univ. Sci. 28 (2016), no. 4, 375–379. DOI: https://doi.org/10.1016/j.jksus.2015.04.008. 10.1016/j.jksus.2015.04.008Search in Google Scholar

[17] S. Al-Omari, q-analogues and properties of the Laplace-type integral operator in the quantum calculus theory, J. Ineq. Appl. 203 (2020), 1–14. DOI: https://doi.org/10.1186/s13660-020-02471-0. 10.1186/s13660-020-02471-0Search in Google Scholar

[18] S. Al-Omari, On a q-Laplace-type integral operator and certain class of series expansion, Math. Meth. Appl. Sci. 240 (2021), no. 10, 1–12. DOI: https://doi.org/10.1002/mma.6002. 10.1002/mma.6002Search in Google Scholar

[19] S. Al-Omari, Estimates and properties of certain q-Mellin transform on generalized q-calculus theory, Adv. Difference Equ. 242 (2021), 1–11. 10.1186/s13662-021-03391-zSearch in Google Scholar

[20] A. Abdeljawad, R. P. Agarwal, E. Karapinar, and P. S. Kumari, Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry 11 (2019), no. 686, 1–18, DOI: https://doi.org/10.3390/sym11050686. 10.3390/sym11050686Search in Google Scholar

[21] B. Alqahtani, H. Aydi, E. Karapinar, and V. Rakocevic, A solution for Volterra fractional integral equations by hybrid contractions, Mathematics 7 (2019), no. 694, 1–10. DOI: https://doi.org/10.3390/math7080694. 10.3390/math7080694Search in Google Scholar

[22] E. Karapinar, T. Abdeljawad, and F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Adv. Difference Equ. 421 (2019), no. 2019, 1–25. 10.1186/s13662-019-2354-3Search in Google Scholar

[23] S. Chandak, D. L. Suthar, S. K. Al-Omari, and S. Gulyaz-Ozyurt, Estimates of classes of generalized special functions and their application in the fractional (k,s)-calculus theory, J. Func. Sp. 2021 (2021), 1–10, DOI: https://doi.org/10.1155/2021/9582879. 10.1155/2021/9582879Search in Google Scholar

[24] N. Khan, T. Usman, M. Aman, S. K. Q. Al-Omari and J. Choi, Integral transforms and probability distributions involving generalized hypergeometric function, Georg. J. Math. 28 (2021), no. 6, 1–12, DOI: https://doi.org/10.1515/gmj-2021-2105. 10.1515/gmj-2021-2105Search in Google Scholar

[25] N. Khan, T. Usman, M. Aman, S. Al-Omari, and S. Araci, Computation of certain integral formulas involving generalized wright function, Adv. Difference Equ. 491 (2020), 1–11, DOI: https://doi.org/10.1186/s13662-020-02948-8. 10.1186/s13662-020-02948-8Search in Google Scholar

[26] S. Jain, P. Agarwal, B. Ahmad, and S. K. Q. Al-Omari, Certain recent fractional integral inequalities associated with the hypergeometric operators, J. King Saud Univ. Sci. 4 (2015), no. 17, 82–86, DOI: https://doi.org/10.1016/j.jksus.2015.04.002. 10.1016/j.jksus.2015.04.002Search in Google Scholar

[27] P. Agarwal, S. Jain, I. O. Kiymaz, M. Chand, and S. K. Q. AL-Omari, Certain sequences of functions involving generalized hypergeometric functions, Math. Scie. Appl. E-Notes 3 (2015), no. 2, 45–53, DOI: https://doi.org/10.36753/mathenot.421329. 10.36753/mathenot.421329Search in Google Scholar

[28] Y. Xiao-Jun, A new integral transform with an application in heat-transfer problem, Thermal Science 20 (2016), no. 3, 677–681. DOI: https://doi.org/10.2298/TSCI16S3677Y. 10.2298/TSCI16S3677YSearch in Google Scholar

[29] S. Al-Omari, Hartley transforms on a certain space of generalized Functions, Georgian Math. J. 20 (2013), no. 30, 415–426. DOI: https://doi.org/10.1515/gmj-2013-0034. 10.1515/gmj-2013-0034Search in Google Scholar

[30] X. Liang, Feng Gaoa, Ya-Nan Gao, Xiao-Jun Yang, Applications of a novel integral transform to partial differentialequations, J. Nonlinear Sci. Appl. 10 (2017), no. 2, 528–534. DOI: https://doi.org/10.22436/jnsa.010.02.18. 10.22436/jnsa.010.02.18Search in Google Scholar

[31] O. Ersoy, A comparative review of real and complex Fourier-related transforms, Proc. IEEE 82 (1994), no. 3, 429–447, DOI: https://doi.org/10.1109/5.272147. 10.1109/5.272147Search in Google Scholar

[32] X.-J. Yang, A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Letters 64 (2017), 193–197, DOI: https://doi.org/10.1016/j.aml.2016.09.011. 10.1016/j.aml.2016.09.011Search in Google Scholar

[33] P. K. Banerji, S. K. Al-Omari, and L. Debnath. Tempered distributional Fourier sine(cosine) transform, Integ. Trans. Spl. Funct. 17 (2006). no. 11, 759–768, DOI: https://doi.org/10.1080/10652460600856534. 10.1080/10652460600856534Search in Google Scholar

[34] A. H. Zemanian, Generalized Integral Transformation, Dover Publications, Inc., New York. First published by Interscience Publishers, New York, 1987. Search in Google Scholar

[35] S. K. Al-Omari and J. F. Al-Omari, Hartley transform for Lp Boehmians and spaces of ultradistributions, Int. Math. Forum 7 (2012), no. 9, 433–443. 10.1155/2012/405368Search in Google Scholar

[36] R. Balasubramanian, S. Ponnusamy, and M. Vuorinen, On hypergeometric functions and function space, J. Comput. Appl. Math. 139 (2002), 299–322. DOI: https://doi.org/10.1016/S0377-0427(01)00417-4. 10.1016/S0377-0427(01)00417-4Search in Google Scholar

[37] R. Balasubramanian, S. Ponnusamy, and D. J. Prabhakaran, Duality techniques for certain integral transforms to be starlike, J. Math. Anal. Appl. 293 (2004), 355–373. DOI: https://doi.org/10.1016/j.jmaa.2004.01.011. 10.1016/j.jmaa.2004.01.011Search in Google Scholar

[38] I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, Inc, New York, 2003. 10.1201/9780203911624Search in Google Scholar

[39] R. Fournier and S. Ruscheweyh, On two extremal problems related to univalent functions, Rocky Mountain J. Math. 24 (1994), no.2, 529–538. DOI: https://doi.org/10.1216/rmjm/1181072416. 10.1216/rmjm/1181072416Search in Google Scholar

[40] L. E. Britvina, A class of integral transforms related to the Fourier cosine convolution, Integ. Trans. Spec. Funct. 16 (2005), no. 5, 379–389, DOI: https://doi.org/10.1080/10652460412331320395. 10.1080/10652460412331320395Search in Google Scholar
Received: 2022-05-21
Revised: 2022-07-31
Accepted: 2022-09-28
Published Online: 2023-01-30

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
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https://doi.org/10.1515/dema-2022-0168
Keywords for this article
convolution; Hadamard product; duality principle; λ-spirallike functions; dual set; Gaussian hypergeometric function
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Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
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  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
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  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
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  26. Feature fusion-based text information mining method for natural scenes
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  28. The study of solutions for several systems of PDDEs with two complex variables
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  32. A dimension expanded preconditioning technique for block two-by-two linear equations
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  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
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  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
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  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
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  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
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  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
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  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
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