JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2021/67392376739237Research ArticleOn Some Relationships of Certain KUniformly Analytic Functions Associated with Mittag-Leffler Functionhttps://orcid.org/0000-0001-7135-7400AttiyaAdel A.12AliEkram E.13https://orcid.org/0000-0003-2907-3353HassanTaher S.12AlbalahiAbeer M.1MurugusundaramoorthyGangadharan1Department of MathematicsCollege of ScienceUniversity of Ha’ilHa’il 81451Saudi Arabiauoh.edu.sa2Department of Mathematics Faculty of ScienceMansoura UniversityMansoura 35516Egyptmans.edu.eg3Department of Mathematics and Computer ScienceFaculty of SciencePort Said UniversityPort Said 42521Egyptpsu.edu.eg2021285202120211442021304202128520212021Copyright © 2021 Adel A. Attiya et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce and investigate several inclusion relationships of new k-uniformly classes of analytic functions defined by the Mittag-Leffler function. Also, integral-preserving properties of these classes associated with the certain integral operator are also obtained.

University of Ha'ilRG-20020
1. Introduction

Let A be the class of analytic functions in the open unit disc U=z:z<1 which in the form (1)fz=z+n=2anzn.

For fz and gzA, we say that the function fz is subordinate to gz, written symbolically as follows: (2)fgorfzgz,

if there exists a Schwarz function wz, which (by definition) is analytic in U with w0=0 and wz<1, zU, such that fz=gwz for all zU. In particular, if the function gz is univalent in U, then we have the following equivalence relation (cf., e.g., [1, 2]; see also ): (3)fzgzf0g0andfUgU.

Let f be as in (1) and hz=z+n=2bnzn, then Hadamard product (or convolution) of fz and hz is given by (4)fhz=z+k=2akbkzkzU.

For ζ,η0,1, we denote by Sζ,Cζ,Kζ,η, and Kζ,η the subclasses of A consisting of all analytic functions which are, respectively, starlike of order ζ, convex of order ζ, close-to-convex of order ζ and type η, and quasiconvex of order ζ and type η in U.

Also, let the subclasses USμ,ζ, UCμ,ζ, USKμ,ζ,η, and UCKμ,ζ,η of Aη0,1<1;μ0 be defined as follows: (5)USμ,ζ=fA:Rzfzfzζ>μzfzfz1,UCμ,ζ=fA:R1+zfzfzζ>μzfzfz,USKμ,ζ,η=fA:hUSμ,ζs.t.Rzfzhzζ>μzfzhz1UCKμ,ζ,η=fA:hUCμ,ζs.t.Rzfzhzζ>μzfzhz1.

We note that (6)US0,ζ=Sζ,UC0,ζ=Cζ,USK0,ζ,η=Kζ,ηandUCK0,ζ,η=Kζ,η0ζ;η<1.

Moreover, let qμ,ζz be an analytic function which maps U onto the conic domain Φμ,ζ=u+iv:u>ku12+v2+ζ such that 1Φμ,ζ defined as follows: (7)qμ,ζz=1+12ζz1zμ=0,1ζ1μ2cos2πcos1μilog1+z1zμ2ζ1μ20<μ<1,1+21ζπ2log1+z1z2μ=1,1ζμ21sinπ2ςμ0uzμdt1t21μ2t2+μ2ζμ21μ>1,where uz=zμ/1μz and ςμ is such that μ=coshπςz/4ςz. By virtue of properties of the conic domain Φμ,ζ (cf., e.g., [4, 5]), we have (8)Rqμ,ζz>μ+ζμ+1.

Making use of the principal of subordination and the definition of qμ,ζz, we may rewrite the subclasses USμ,ζ, UCμ,ζ, USKμ,ζ,η, and UCKμ,ζ,η as follows: (9)USμ,ζ=fA:zfzfzqμ,ζz,UCμ,ζ=fA:1+zfzfzqμ,ζz,USKμ,ζ,η=fA:hUSμ,ηs.t.zfzhzqμ,ζz

and (10)UCKμ,ζ,η=fA:hUCμ,ζs.t.zfzhzqμ,ζz.

Attiya  introduced the operator Hα,βγ,kf, where Hα,βγ,kf: AA is defined by (11)Hα,βγ,kf=μα,βγ,kfzzU,with β,γ,Reα>max0,Rek1 and Rek>0. Also, Reα=0 when Rek=1; β0. Here, μα,βγ,k is the generalized Mittag–Leffler function defined by , see also , and the symbol () denotes the Hadamard product.

Due to the importance of the Mittag–Leffler function, it is involved in many problems in natural and applied science. A detailed investigation of the Mittag–Leffler function has been studied by many authors (see, e.g., ).

Attiya  noted that (12)Hα,βγ,kfz=z+n=2Γγ+nkΓα+βΓγ+kΓβ+αnn!anzn.

Also, Attiya  showed that (13)zHα,βγ,kfz=γ+kkHα,βγ+1,kfzγkHα,βγ,kfz,

and (14)zHα,β+1γ,kfz=α+βαHα,βγ,kfzβαHα,β+1γ,kfz.

Next, by using the operator Hα,βγ,kf, we introduce the following subclasses of analytic functions in U(15)USβγμ,ζ=fA:Hα,βγ,kfzUSμ,ζ,UCβγμ,ζ=fA:Hα,βγ,kfzUCμ,ζ,USKβγμ,ζ,η=fA:Hα,βγ,kfzUSKμ,ζ,η,UCKβγμ,ζ,η=fA:Hα,βγ,kfzUCKμ,ζ,η,where β,γ,Rα>max0,Rk1and Rk>0. Also, Rα=0 when Rk=1; β0.

Also, we note that (16)fzUCβγμ,ζzfzUSβγμ,ζ,(17)fzUCKβγμ,ζ,ηzfzUSKβγμ,ζ,η.

In this paper, we introduce several inclusion properties of the classes USβγμ,ζ, UCβγμ,ζ, USKβγμ,ζ,η, and UCKβγμ,ζ,η. Also, integral-preserving properties of these classes associated with generalized Libera integral operator are also obtained.

2. Inclusion Properties Associated with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M88"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">H</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">α</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">β</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">γ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">k</mml:mi></mml:mrow></mml:msubsup><mml:mi mathvariant="bold-italic">f</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="bold-italic">z</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>Lemma 1 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

If hz is convex univalent in U with h0=1 and Rξhz+ζ>0ζ. Let pz be analytic in U with p0=1 which satisfy the following subordination relation (18)pz+zpzξpz+ζhz,then (19)pzhz.

Lemma 2 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

If hz is convex univalent in U and let w be analytic in U with Rwz0. Let pz be analytic in U and p0=h0 which satisfy the following subordination relation (20)pz+wzzpzhz,then (21)pzhz.

Theorem 3.

If Rγ/k>μ+ζ/μ+1, then USβγ+1μ,ζUSβγμ,ζ.

Proof.

Let fzUSβγ+1μ,ζ, put (22)pz=zHα,βγ,kfzHα,βγ,kfzzU,we note that pz is analytic in U and p0=1. From (13) and (22), we have (23)Hα,βγ+1,kfzHα,βγ,kfz=kγ+kpz+γk.

Differentiating (23) with respect to z, we obtain (24)zHα,βγ+1,kfzHα,βγ+1,kfz=pz+zpzpz+γ/k.

From the above relation and using (7), we may write (25)pz+zpzpz+γ/kqμ,ζzzU.

Since Rqμ,ζz>μ+ζ/μ+1, we see that (26)Rqμ,ζz+γk>0zU.

Applying Lemma 1, it follows that pzqμ,ζz, that is, fzUSβγμ,ζ.

Using the same technique in Theorem 3 with relation (14), we have the following theorem.

Theorem 4.

If Rα/β>μ+ζ/μ+1, then USβγμ,ζUSβ+1γμ,ζ.

Theorem 5.

If Rγ/k>μ+ζ/μ+1, then UCβγ+1μ,ζUCβγμ,ζ.

Proof.

Applying Theorem 3 and relation (16), we observe that (27)fzUCβγ+1μ,ζzfzUSβγ+1μ,ζzfzUSβγμ,ζfzUCβγμ,ζ,which evidently proves Theorem 5.

Similarly, we can prove the following theorem.

Theorem 6.

If Rα/β>μ+ζ/μ+1, then UCβγμ,ζUCβ+1γμ,ζ.

Theorem 7.

If Rγ/k>μ+ζ/μ+1, then USKβγ+1μ,ζ,ηUSKβγμ,ζ,η.

Proof.

Let fzUSKβγ+1μ,ζ,η. Then, there exists a function rzUSμ,ζ such that (28)zHα,βγ+1,kfzrzqμ,ζz.

We can choose the function hz such that Hα,βγ+1,khz=rz. Then, hzUSβγ+1μ,ζ and (29)zHα,βγ+1,kfzHα,βγ+1,khzqμ,ζz.

Now, let (30)pz=zHα,βγ,kfzHα,βγ,khz,where pz is analytic in U with p0=1. Since hzUSβγ+1μ,ζ, by Theorem 3, we know that hzUSβγμ,ζ. Let (31)tz=zHα,βγ,khzHα,βγ,khzzU,where tz is analytic in U with Rtz>μ+ζ/μ+1. Also, from(30), we note that (32)zHα,βγ,kfz=Hα,βγ,kzfz=Hα,βγ,khzpz.

Differentiating both sides of (32) with respect to z, we obtain (33)zHα,βγ,kzfzHα,βγ,khz=zHα,βγ,khzHα,βγ,khzpz+zpz=tzpz+zpz.

Now, using (13) and (33), we obtain (34)zHα,βγ,kfzHα,βγ,khz=Hα,βγ+1,kzfzHα,βγ+1,khz=zHα,βγ,kzfz+γ/kHα,βγ,kzfzzHα,βγ,khz+γ/kHα,βγ,khz=zHα,βγ,kzfz/Hα,βγ,khz+γ/kzHα,βγ,kfz/Hα,βγ,khzzHα,βγ,khz/Hα,βγ,khz+γ/k=tzpz+zpz+γ/kpztz+γ/k=pz+zpztz+γ/k.

Since Rγ/k>μ+ζ/μ+1, we see that (34)Rtz+γk>0zU.

Hence, applying Lemma 2, we can show that pzqμ,ζz, so that fzUSKβγμ,ζ,η. This completes the proof of Theorem 7.

Similarly, we can prove the following theorem.

Theorem 8.

If Rα/β>μ+ζ/μ+1, then USKβγμ,ζ,ηUSKβ+1γμ,ζ,η.

We can also prove Theorem 9 by using Theorem 7 and relation (17).

Theorem 9.

If Rγ/k>μ+ζ/μ+1, then UCKβγ+1μ,ζ,ηUCKβγμ,ζ,η.

Also, we obtain the following theorem.

Theorem 10.

If Rα/β>μ+ζ/μ+1, then UCKβγμ,ζ,ηUCKβ+1γμ,ζ,η.

Now, we obtain squeeze theorems for inclusion by combining the above theorems as follows:

Combining both theorems 3 and 4, we have the following corollary.

Corollary 11.

If μ+ζ/μ+1>minRγ/k,Rα/β, then (36)USβγ+1μ,ζUSβγμ,ζUSβ+1γμ,ζ.

Combining both theorems 5 and 6, we have the following corollary.

Corollary 12.

If μ+ζ/μ+1>minRγ/k,Rα/β, then (37)UCβγ+1μ,ζUCβγμ,ζUCβ+1γμ,ζ.

Combining both theorems 7 and 8, we have the following corollary.

Corollary 13.

If μ+ζ/μ+1>minRγ/k,Rα/β, then (38)USKβγ+1μ,ζ,ηUSKβγμ,ζ,ηUSKβ+1γμ,ζ,η.

Combining both theorems 9 and 10, we have the following corollary.

Corollary 14.

If μ+ζ/μ+1>minRγ/k,Rα/β, then (39)UCKβγ+1μ,ζ,ηUCKβγμ,ζ,ηUCKβ+1γμ,ζ,η.

3. Integral Preserving Properties Associated with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M173"><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">F</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">δ</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

The generalized Libera integral operator Fδ (see , also, see related topics ) is defined by (40)Fδfz=δ+1zδ0ztδ1ftdt,where fzA and δ>1.

Theorem 15.

Let δ>μ+ζ/μ+1. If fUSβγμ,ζ, then FδfUSβγμ,ζ.

Proof.

Let fUSβγμ,ζ and set (41)pz=zHα,βγ,kFδfzHα,βγ,kFδfzzU,where pz is analytic in U with p0=1. From definition of Hα,βγ,kf and (40), we have (42)zHα,βγ,kFδfz=δ+1Hα,βγ,kfzδHα,βγ,kFδfz.

Then, by using (41) and (42), we obtain (43)δ+1Hα,βγ,kfzHα,βγ,kFδfz=pz+δ.

Taking the logarithmic differentiation on both sides of (43) and simple calculations, we have (44)pz+zpzpz+δ=zHα,βγ,kfzHα,βγ,kfzqμ,ζz.

Since Rqμ,ζ+δ>μ+ζ/μ+1+δ>0, by virtue of Lemma 1, we conclude that pzqμ,ζz in U, which implies that FδfUSβγμ,ζ.

Theorem 16.

Let δ>μ+ζ/μ+1. If fUCβγμ,ζ, then FδfUCβγμ,ζ.

Proof.

By applying Theorem 15, it follows that (45)fzUCβγμ,ζzfzUSβγμ,ζFδzfzUSβγμ,ζzFδfzUSβγμ,ζFδfzUCβγμ,ζ,which proves Theorem 16.

Theorem 17.

Let δ>μ+ζ/μ+1. If fUSKβγμ,ζ,η, then FδfUSKβγμ,ζ,η.

Proof.

Let fzUSKβγμ,ζ,η. Then, there exists a function hzUSβγμ,ζ such that (46)zHα,βγ,kfzHα,βγ,khzqμ,ζz.

Thus, we set (47)pz=zHα,βγ,kFδfzHα,βγ,kFδhzzU,where pz is analytic in U with p0=1. Since hzUSβγμ,ζ, we see from Theorem 15 that FδhUSβγμ,ζ. Let (48)tz=zHα,βγ,kFδhzHα,βγ,kFδhz,where tz is analytic in U with Rtz>μ+ζ/μ+1. Using (47), we have (49)Hα,βγ,kzFδfz=Hα,βγ,kFδhzpz.

Differentiating both sides of (49) with respect to z and simple calculations, we obtain (50)zHα,βγ,kzFδfzHα,βγ,kFδhz=zHα,βγ,kFδhzHα,βγ,kFδhzpz+zpz=tzpz+zpz.

Now, using the identity (42) and (50), we obtain (51)zHα,βγ,kfzHα,βγ,khz=Hα,βγ,kzfzHα,βγ,khz=zHα,βγ,kzFδfz+δHα,βγ,kzFδfzzHα,βγ,kFδhz+δHα,βγ,kFδhz=zHα,βγ,kzFδfz/Hα,βγ,kFδhz+δzHα,βγ,kFδfz/Hα,βγ,kFδhzzHα,βγ,kFδhz/Hα,βγ,kFδhz+δ=tzpz+zpz+δpztz+δ=pz+zpztz+δ.

Since δ>μ+ζ/μ+1 and Rtz>μ+ζ/μ+1, we see that (52)Rtz+δ>0zU.

Applying Lemma 2 into relation (51), it follows that pzqμ,ζz, which is FδfUSKβγμ,ζ,η.

We can deduce the integral-preserving property asserted by 18 by using Theorem 17 and relation (17).

Theorem 18.

Let δ>μ+ζ/μ+1. If fUCKβγμ,ζ,η, then FδfUCKβγμ,ζ,η.

Data Availability

All data are available in this paper.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

The authors contributed equally to the writing of this paper. All authors approved the final version of the manuscript.

Acknowledgments

This research has been funded by Scientific Research Deanship at the University of Ha'il, Saudi Arabia, through project number RG-20020.

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