Inclusion relations for certain families of integral operators associated with conic regions

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Citation for this paper:

Mahmood, S., Khan, I., Srivastava, H.M. & Malik, S.N. (2019). Inclusion relations

for certain families of integral operators associated with conic regions. Journal of

Inequalitites and Applications, 2019:59.

https://doi.org/10.1186/s13660-019-2015-9

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Inclusion relations for certain families of integral operators associated with conic

regions

Shahid Mahmood, Imran Khan, Hari Mohan Srivastava and Sarfraz Nawaz Malik

March 2019

© The Author(s) 2019. This article is distributed under the terms of the Creative

Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/ ),which permits unrestricted use,

distribution, and reproduction in any medium, provided you give appropriate credit

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This article was originally published at:

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R E S E A R C H

Open Access

Inclusion relations for certain families of

integral operators associated with conic

regions

Shahid Mahmood

1*

_{, Imran Khan}

1

_{, Hari Mohan Srivastava}

2,3

_{and Sarfraz Nawaz Malik}

4

*_{Correspondence:}

shahidmahmood757@gmail.com 1_{Department of Mechanical}

Engineering, Sarhad University of Science & I. T, Peshawar, Pakistan Full list of author information is available at the end of the article

Abstract

In this work, we introduce certain subclasses of analytic functions involving the integral operators that generalize the class of uniformly starlike, convex, and close-to-convex functions with respect to symmetric points. We then establish various inclusion relations for these newly deﬁned classes.

MSC: 30C45; 30C50

Keywords: Sakaguchi functions; Schwarz function; Subordination; Functions with

positive real parts; Analytic functions; Conic domain; Uniformly starlike; Integral operators; Symmetrical points

1 Introduction

LetA be the class of functions

f(z) = z +

∞

n=2

anzn (1.1)

analytic in the open unit disc A ={z ∈ C : |z| < 1}, and letS be the class of functions in

A that are univalent in A. Also let S∗_,_{C, K, and C}∗ _{be the subclasses of}_{A consisting of}

all functions that are starlike, convex, close-to-convex, and quasiconvex, respectively; for details, see [1].

Let f and g be analytic in A. We say that f is subordinate to g, written as f (z)≺ g(z), if there exists a Schwarz function w that is analytic in A with w(0) = 0 and|w(z)| < 1 (z ∈ A) and such that f (z) = g(w(z)). In particular, when g is univalent, then such a subordination is equivalent to f (0) = g(0) and f (A)⊆ g(A); see [1].

Two points A and Aare said to be symmetrical with respect to M if M is the midpoint of the line segment AA. Sakaguchi [2] introduced and studied the class S_s∗ of starlike functions with respect to symmetrical points z and –z belonging to the open unit disc A. The classS_s∗ includes the classes of convex and odd starlike functions with respect to the origin. It was shown [2] that a necessary and suﬃcient condition for f (z)∈S_s∗ to be univalent and starlike with respect to symmetrical points in A is that

2zf(z)

f(z) – f (–z)∈P, z ∈ A.

©The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Das and Singh [3] deﬁned the classesCsof convex functions with respect to symmetrical

points and showed that a necessary and suﬃcient condition for f (z)∈Csis that

2(zf(z))

(f (z) – f (–z)) ∈P, z ∈ A.

It is also well known [3] that f (z)∈Csif and only if zf (z)∈Ss∗.

The classes k –CV and k – ST with k ≥ 0 denote the famous classes of k-uniformly convex and k-starlike functions, respectively, introduced by Kanas and Wisniowska, re-spectively. For some details see [4–7].

Consider the domain

Ωk=

u+ iv; u > k(u – 1)2_{+ v}2_. _(1.2)

For ﬁxed k, Ωkrepresents the conic region bounded successively by the imaginary axis

(k = 0), the right branch of a hyperbola (0 < k < 1), a parabola (k = 1), and an ellipse (k > 1). This domain was studied by Kanas [4–6]. The function pk with pk(0) = 1 and pk(0) > 0

plays the role of extremal and is given by

pk(z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1+z 1–z, k= 0, 1 +_π22(log 1+√z 1–√z) 2_, _k_{= 1,}

1 +_1–k22sinh2[(π2arccosk) arc tanh

√ z], 0 < k < 1, 1 +_k21_–1sin[_2R(t)π u_√(z) t 0 √ 1 1–x2√_1–(tx)2dx] + 1 k2_–1, k> 1, (1.3) with u(z) = z– √ t

1–√tz, t∈ (0, 1), z ∈ E, and t chosen such that k = cosh( πR(t)

4R(t)), where R(t) is

Leg-endre’s complete elliptic integral of the ﬁrst kind, and R(t) is the complementary integral of R(t) (see [5,6]). LetPpk denote the class of all functions p(z) that are analytic in E with

p(0) = 1 and p(z)≺ pk(z) for z∈ E. Clearly, we can see thatPpk⊂P, where P is the class of functions with positive real parts (see [1]). More precisely,

Ppk⊂P k 1 + k ⊂P.

For more detail regarding conic domains and related classes, see [4–6,8–11].

Recently, Noor [12] deﬁned the classes k –STs, k –UCVs, and k –UKsof k-uniformly

starlike, convex, and close to convex functions with respect to symmetrical points and studied various interesting properties for these classes.

We consider the following one-parameter families of integral operators:

Iα βf(z) = (β + 1)α Γ(α)zβ z 0 tβ–1 logz t α–1 f(t) dt, (1.4) Lα_βf(z) = α+ β β α zβ z 0 tβ–1 1 –t z α–1 f(t) dt, (1.5) and Jβf(z) = β+ 1 zβ z 0 tβ–1f(t) dt, (1.6)

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where α≥ 0, β > –1, and Γ is the familiar gamma function. We note that Jβ:A → A

deﬁned by (1.6) is the generalized Bernardi operator introduced in [13] for β = 1, 2, 3, . . . , and for any real number β > –1, this operator was studied by Owa and Srivastava [14,15]. For the operators Lα

βandIβα, we refer to [16,17]. Also, for α = 1, we see that

Jβf(z) = L1βf(z) =Iβ1f(z).

We can represent these operators as follows:

Iα βf(z) = z + ∞ n=2 β+ 1 β+ n α anzn = z+ ∞ n=2 β+ 1 β+ n α zn ∗ f (z), (1.7) Lα_βf(z) = z + ∞ n=2 Γ(β + n)Γ (α + β + 1) Γ(α + β + n)Γ (β + 1)anz n = α+ β β z2F1(1, β; α + β; z)∗ f (z), (1.8) and Jβf(z) = z + ∞ n=2 β+ 1 β+ n anzn, (1.9)

where2F1denotes the Gaussian hypergeometric function, and the symbol∗ stands for the

convolution (Hadamard product).

By (1.7) and (1.8) we can easily derive the identities

zIβαf(z) = (β + 1)Iβα–1f(z) – βI α βf(z) (1.10) and zLα_βf(z)= (α + β)Lαβ–1f(z) – (α + β – 1)L α βf(z), (1.11)

where α≥ 1 and β > –1. From (1.10) we have 1 1 + βp(z) + β 1 + β =I α–1 β f(z) Iα βf(z) with p(z) =z(I α βf(z)) Iα βf(z) .

With the help of these integral operators, we now deﬁne the following classes.

Deﬁnition 1.1 Let f (z)∈A. Then f (z) ∈ k – STs(α, β), α≥ 0, β > –1, ifIβαf(z)∈ k –STs

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Deﬁnition 1.2 Let f (z)∈A. Then f (z) ∈ k – ST∗_s(α, β), α≥ 0, β > –1, if Lα

βf(z)∈ k –STs

in A.

Deﬁnition 1.3 Let f (z)∈A. Then f (z) ∈ k – UKs(α, β), α≥ 0, β > –1, ifIβαf(z)∈ k –UKs

in A.

Deﬁnition 1.4 Let f (z)∈A. Then f (z) ∈ k – UK∗_s(α, β), α≥ 0, β > –1, if Lα

βf(z)∈ k –UKs

in A.

2 A set of lemmas

In this section, we give the following lemmas, which will be used in our investigation.

Lemma 2.1 ([4]) Let k≥ 0, and let β1, γ ∈ C be such that β1= 0 and Re{β_k₊₁1k + γ} > 0.

Suppose that p(z) is analytic in A with p(0) = 1 and satisﬁes p(z) + zp _(z) β1p(z) + γ ≺ pk(z) (2.1)

and that q(z) is an analytic function satisfying

q(z) + zq

_(z)

β1q(z) + γ

= pk(z). (2.2)

Then q(z) is univalent, p(z)≺ q(z) ≺ pk(z), and q(z) is the best dominant of (2.1) given as

q(z) = β1 1 0 tβ1+γ –1_exp tz z pk(u) – 1 u du dt –1 – γ β1 . (2.3)

Lemma 2.2([18]) Let λ, ρ∈ C be such that λ = 0, and let φ(z) ∈A be convex and univalent

inU with Re{λφ(z) + ρ} > 0 (z ∈ U). Also, let q(z) ∈A and q(z) ≺ φ(z). If p(z) is analytic in

U with p(0) = 1 and satisﬁes p(z) + zp _(z) λq(z) + ρ ≺ φ(z), (2.4) then p(z)≺ φ(z).

3 The main results and their consequences

Our ﬁrst main result is stated as the following:

Theorem 3.1 Let f(z)∈ k –STs(α, β). Then the odd function

ψ(z) =1 2

f(z) – f (–z)∈ k –ST (α, β).

Proof Note that

Iα βψ(z) = 1 2 Iα βf(z) –I α βf(–z) .

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We want to show thatIα

βψ(z)∈ k –ST . Now, for f (z) ∈ k – STs(α, β), this implies that

Iα βf(z)∈ k –STs. Then, for z∈ A, z(Iα βψ(z)) Iα βψ(z) =1 2 _2z(_Iα βf(z)) Iα βf(z) –I α βf(–z) + 2(–z)(I α βf(–z)) Iα βf(–z) –I α βf(z) =1 2 h1(z) + h2(z) = h(z).

and hi(z)≺ pk(z), i = 1, 2. This implies that h(z)≺ pk(z) in A, and thereforeIβαψ(z)∈ k –

ST . Consequently, ψ(z) ∈ k – ST (α, β) in A. Similarly, we can prove that if f (z)∈ k –ST∗_s(α, β), then

φ(z) =1 2

f(z) – f (–z)∈ k –ST∗(α, β).

Taking α = 0, we obtain the following result proved by Noor [12].

Corollary 3.2 Let f(z)∈ k –STs. Then the odd function

ψ(z) =1 2

f(z) – f (–z)∈ k –ST .

Note that, for k = α = 0, the function ψ(z) =1

2[f (z) – f (–z)] is a starlike function in A;

see [2].

Theorem 3.3 Let α≥ 2 and β > –1. Then k –ST (α – 1, β) ⊂ k – ST (α, β). Proof Let f (z)∈ k –ST (α – 1, β) and set

p(z) =z(I α βf(z)) Iα βf(z) . (3.1)

Note that p(z) is analytic in A with p(0) = 1. From (3.1) and identity (1.10) we have

Iα–1 β f(z) Iα βf(z) = (1 – γ )p(z) + γ (3.2) with γ = β β+ 1. (3.3)

Logarithmic diﬀerentiation of (3.2) yields

z(Iα–1 β f(z)) Iα–1 β f(z) = p(z) + (1 – γ )zp _(z) (1 – γ )zp(z) + γ ,

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and thus it follows that p(z) + zp _(z) zp(z) + β ≺ pk(z).

Using Lemma2.1, we have

p(z)≺ q(z) ≺ pk(z) with q(z) = 1 0 tβexp tz z pk(u) – 1 u du dt –1 – β.

This proves that f (z)∈ k –ST (α, β) in A, and the proof is complete.

Theorem 3.4 Let α≥ 2 and β > –1. Then k –ST∗(α – 1, β)⊂ k –ST∗(α, β).

Proof Let

z(Lα βf(z))

Lα_βf(z) = h(z), (3.4)

where h(z) is analytic in A with h(0) = 1. From (3.4) and identity (1.11) we get

1 α+ β z(Lα βf(z)) Lα_βf(z) + 1 – 1 α+ β =L α–1 β f(z) Lα_βf(z) . (3.5)

Logarithmic diﬀerentiation of (3.5), together with (3.4), gives us

z(Lα–1 β f(z)) Lα_β–1f(z) = h(z) + 1 α+βzh(z) 1 α+βh(z) + α+β–1 α+β = h(z) + zh _(z) h(z) + α + β – 1. Since f (z)∈ k –ST∗(α – 1, β), it follows that

h(z) + zh

_(z)

h(z) + α + β – 1≺ pk(z). Applying Lemma2.1, we have

h(z)≺ pk(z).

This proves our result.

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Proof Let f (z)∈ k –STs(α – 1, β). Then, using Theorems3.1and3.3, we have

ψ(z) =f(z) – f (–z)

2 ∈ k –ST (α – 1, β) ⊂ k – ST (α, β).

From this it easily follows that f (z)∈ k –STs(α, β), and this completes the proof.

A similar result for the class k –ST∗_s(α, β) can be easily proved.

Theorem 3.6 Let α≥ 1 and β > 0. Then k –UKs(α – 1, β)⊂ k –UKs(α, β).

Proof Let f (z)∈ k –UKs(α – 1, β). Then there exists g(z)∈ k –STs(α – 1, β) such that

2z(Iβα–1f(z)) Iα–1 β g(z) –I α–1 β g(–z) =z(I α–1 β f(z)) Iα–1 β ψ(z) ∈ P, where ψ(z) =I α–1 β g(z)–Iαβ–1g(–z) 2 ∈ k –ST (α – 1, β) ⊂ k – ST (α, β) in A. Let us set z(Iα βf(z)) Iα βψ(z) = p(z), (3.6)

where p(z) is analytic in A with p(0) = 1. Then by (3.6) and identity (1.10) we get

Iα–1 β ψ(z) Iα βψ(z) = (1 – γ )p0(z) + γ , where p0(z) = z(Iα βψ(z)) Iα

βψ(z) , and γ is given by (3.3). Now by simple computations we obtain

z(Iα–1 β f(z)) zIα–1 β ψ(z) = z(I α–1 β f(z)) Iα βψ(z)[(1 – γ )p0(z) + γ ] = z[(z(I α βf(z)))] + βz(I α βf(z)) (β + 1)Iα βψ(z)[(1 – γ )p0(z) + γ ] = βp(z) + p(z)p0(z) + zp _(z) (β + 1)[(1 –_1+ββ )p0(z) +_1+ββ ] =βp(z) + p(z)p0(z) + zp _(z) p0(z) + β = p(z) + zp _(z) p0(z) + β .

Since f (z)∈ k –UKs(α – 1, β), it follows that

p(z) + zp

_(z)

p0(z) + β

∈P in A.

Applying Lemma.2.2, we have p(z)∈P in A. This proves f (z) ∈ k – UKs(α, β) in A.

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Theorem 3.7 Let α≥ 1 and β > 0. Then k –UK∗(α – 1, β)⊂ k –UK∗(α, β).

Theorem 3.8 Let f(z)∈ k –STs(α, β) in A. Then

Re _z₍_Iα–1 β f(z)) Iα–1 β ϕ(z) > 0 for|z| < R(β, γ0), where R(β, γ0) = (1 + β) (2 – γ0) + (2 – γ0)2+ (1 + β)(β + 2γ0– 1) with γ0= k k+ 1. (3.7)

Proof Let f (z)∈ k –STs(α, β). Then

ϕ(z) =f(z) – f (–z) 2 ∈ k –ST (α, β), and hence z(Iα βf(z)) Iα βϕ(z) ∈P(pk)⊂P(γ0),

where γ0is given by (3.7). Let

z(Iα βf(z)) Iα βϕ(z) = h(z), h(z)∈P(γ0), = (1 – γ0)h0(z) + γ0, h0(z)∈P. (3.8)

Then, proceeding as in Theorem3.5, we have

z(Iα–1 β f(z)) Iα–1 β ϕ(z) = h(z) + zh _(z) p(z) + β, (3.9) where p(z) =z(I α βϕ(z)) Iα

βϕ(z) ∈P(γ ). Using (3.8) and p(z) = (1 – γ0)p0(z) + γ0in (3.9), we have

z(Iα–1 β f(z)) Iα–1 β ϕ(z) = (1 – γ0)h0(z) + γ0+ (1 – γ0)zh0(z) (1 – γ0)p0(z) + γ0+ β

with h0(z)∈P, p0(z)∈P, that is,

1 1 – γ0 _z₍_Iα–1 β f(z)) Iα–1 β ϕ(z) – γ0 = h0(z) + zh₀(z) (1 – γ0)p0(z) + γ0+ β .

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Using the distortion result for the classP, we obtain Re 1 1 – γ0 _z₍_Iα–1 β f(z)) Iα–1 β ϕ(z) – γ0 ≥ Reh0(z) 1 – 2r 1–r2 (1 – γ0)1–r_1+r+ (γ0+ β) = Reh0(z) 1 – 2r (1 – γ0)(1 + r)2+ (1 – r2)(γ0+ β) . (3.10)

Right-hand side of (3.10) is greater than or equal to zero for|z| < R(β, γ0), where R(β, γ0)

is the least positive root of the equation

T(r) := (1 – β – 2γ0)r2– 2(2 – γ0)r + (1 + β) = 0, that is, R(β, γ0) = 2(2 – γ0) – 4(2 – γ0)2+ 4(1 + β)(β + 2γ0– 1) 2(1 – β – 2γ0) = (1 + β) (2 – γ0) + (2 – γ0)2+ (1 + β)(β + 2γ0– 1) .

The proof is completed.

Particular Cases

(i) For β = 0 and γ0=_kk₊₁= 0(i.e., k = 0), we have f (z)∈Ss∗(α, 0)(ψ∈S∗(α, 0)) and

R(0, 0) = 1 2 +√3. (ii) For k = 1 and β = 0,

R 0,1 2 =1 3. (iii) For k = 1 and β = 1,

R 1,1 2 = 4 4 +√17. Theorem 3.9 Let Lα βf(z)∈ k –ST . Then Lα_β–1f(z)∈S∗(γ0), γ0= k k+ 1 for|z| < R1, where R1(α, β, γ0) = α+ β 2 – γ0+ (2 – γ0)2+ (α + β)(2γ0+ α + β – 2) .

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Proof Since Lα

βf(z)∈ k –ST , we have

z(Lα βf(z))

Lα_βf(z) = h(z), h(z)≺ pk(z)

in A. With a similar argument as in Theorem3.5, we have

z(Lα–1 β f(z)) Lα_β–1f(z) = h(z) + zh(z) h(z) + α + β – 1, that is, Re 1 1 – γ0 _z_(Lα–1 β f(z)) Lα_β–1f(z) – γ0 = Re h0(z) + zh₀(z) (1 – γ0)h0(z) + (γ0+ α + β – 1) ≥ Reh0(z) 1 – 2r 1–r2 (1 – γ0)1–r_1+r+ (γ0+ α + β – 1) , (3.11) where h(z) = (1 – γ0)h0(z) + γ0, h0∈P, γ0= k k+ 1.

The right-hand side of (3.11) is greater than or equal to zero for|z| < R1, where R1is the

least positive root of the equation

T(r) := (2 – 2γ0– α – β)r2– 2(2 – γ0)r + α + β = 0, that is, R1(α, β, γ0) = 2 – γ0– (2 – γ0)2+ (α + β)(2γ0+ α + β – 2) 2(2 – α – β – 2γ0) = α+ β 2 – γ0+ (2 – γ0)2+ (α + β)(2γ0+ α + β – 2) .

This completes the proof.

4 Conclusion

In this paper, we have deﬁned some new classes of analytic functions involving integral op-erators. We have shown that these classes generalize the well-known classes, and already existing results can be obtained as a particular cases of our results. Inclusion relations of these classes are also a signiﬁcant part of our work. We believe that the work presented in this paper will give researchers a new direction and will motivate them to explore more interesting facts on similar lines.

Acknowledgements

The authors would like to thank the reviewers of this paper for his/her valuable comments on the earlier version of the paper. They would also like to acknowledge Prof. Dr. Salim ur Rehman, V.C. Sarhad University of Science & I. T, for providing excellent research and academic environment.

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Funding

Sarhad University of Science & I. T Peshawar.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results, and they read and approved the ﬁnal manuscript.

Author details

1_{Department of Mechanical Engineering, Sarhad University of Science & I. T, Peshawar, Pakistan.}2_{Department of}

Mathematics and Statistics, University of Victoria, Victoria, Canada.3_{Department of Medical Research, China Medical}

University Hospital, China Medical University, Taichung, Taiwan, Republic of China.4_{Department of Mathematics,}

COMSATS University Islamabad, Wah Campus, Pakistan.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 14 November 2018 Accepted: 27 February 2019

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