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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

and Sukhjit Singh vol. 10, iss. 4, art. 113, 2009

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AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS

SUKHWINDER SINGH SUSHMA GUPTA AND SUKHJIT SINGH

Department of Applied Sciences Department of Mathematics B.B.S.B. Engineering College S.L.I.E.T. Longowal-148 106 Fatehgarh Sahib-140 407 Punjab, India

Punjab, India EMail:sushmagupta1@yahoo.com

EMail:ssbilling@gmail.com EMail:sukhjit_d@yahoo.com

Received: 03 May, 2009

Accepted: 05 November, 2009 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C80, 30C45.

Key words: Analytic Function, Univalent function, Starlike function, Differential subordina- tion.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

and Sukhjit Singh vol. 10, iss. 4, art. 113, 2009

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Close Abstract: We show that forα(0,2],iff ∈ Awithf0(z)6= 0, zE,satisfies the

condition

(1α)f0(z) +α

1 + zf00(z) f0(z)

F(z),

thenfis univalent inE,whereFis the conformal mapping of the unit disk EwithF(0) = 1and

F(E) =C\n

wC:<w=α, |=w| ≥p

α(2α)o . Our result extends the region of variability of the differential operator

(1α)f0(z) +α

1 + zf00(z) f0(z)

, implying univalence off ∈ AinE,for0< α2.

Acknowledgment: The authors are thankful to the referee for valuable comments.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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Contents

1 Introduction and Preliminaries 4

2 Main Result 7

3 Applications to Univalent Functions 9

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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1. Introduction and Preliminaries

Let H be the class of functions analytic in E = {z : |z| < 1} and for a ∈ C (set of complex numbers) and n ∈ N (set of natural numbers), let H[a, n] be the subclass ofHconsisting of functions of the formf(z) =a+anzn+an+1zn+1+· · ·. Let A be the class of functions f, analytic in E and normalized by the conditions f(0) =f0(0)−1 = 0.

Letf be analytic inE, g analytic and univalent inEandf(0) = g(0). Then, by the symbolf(z)≺g(z)(f subordinate tog) inE, we shall meanf(E)⊂g(E).

Letψ :C×C→Cbe an analytic function,pbe an analytic function inE, with (p(z), zp0(z))∈C×Cfor allz ∈ Eandhbe univalent inE, then the functionpis said to satisfy first order differential subordination if

(1.1) ψ(p(z), zp0(z))≺h(z), ψ(p(0),0) =h(0).

A univalent functionqis called a dominant of the differential subordination (1.1) if p(0) =q(0)andp≺qfor allpsatisfying (1.1). A dominantq˜that satisfiesq˜≺qfor all dominantsqof (1.1), is said to be the best dominant of (1.1). The best dominant is unique up to a rotation ofE.

Denote byS(α)andK(α), respectively, the classes of starlike functions of order αand convex functions of orderα, which are analytically defined as follows:

S(α) =

f ∈ A :<

zf0(z) f(z)

> α, z ∈E, 0≤α <1

,

and

K(α) =

f ∈ A:<

1 + zf00(z) f0(z)

> α, z ∈E, 0≤α <1

.

We write S = S(0), the class of univalent starlike convex functions (w.r.t. the origin) andK(0) =K, the class of univalent convex functions.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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A functionf ∈ Ais said to be close-to-convex if there is a real numberα,−π/2<

α < π/2, and a convex functiong(not necessarily normalized) such that

<

ef0(z) g0(z)

>0, z ∈E.

It is well-known that every close-to-convex function is univalent. In 1934/35, Noshiro [4] and Warchawski [8] obtained a simple but interesting criterion for univalence of analytic functions. They proved that if an analytic functionf satisfies the condition

<f0(z)>0for allz inE, thenf is close-to-convex and hence univalent inE. Letφbe analytic in a domain containingf(E), φ(0) = 0and<φ0(0) >0, then, the functionf ∈ Ais said to beφ-like inEif

<

zf0(z) φ(f(z))

>0, z ∈E.

This concept was introduced by Brickman [2]. He proved that an analytic function f ∈ A is univalent if and only if f is φ-like for some φ. Later, Ruscheweyh [5]

investigated the following general class ofφ-like functions:

Letφbe analytic in a domain containingf(E),φ(0) = 0, φ0(0) = 1andφ(w)6= 0 for w ∈ f(E)\ {0}. Then the function f ∈ A is called φ-like with respect to a univalent functionq, q(0) = 1,if

zf0(z)

φ(f(z)) ≺q(z), z ∈E.

LetHα(β)denote the class of functionsf ∈ Awhich satisfy the condition

<

(1−α)f0(z) +α

1 + zf00(z) f0(z)

> β, z ∈E,

where α and β are pre-assigned real numbers. Al-Amiri and Reade [1], in 1975, have shown that forα ≤ 0and for α = 1, the functions inHα(0) are univalent in

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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E. In 2005, Singh, Singh and Gupta [7] proved that for0 < α < 1, the functions inHα(α)are also univalent. In 2007, Singh, Gupta and Singh [6] proved that the functions inHα(β) satisfy the differential inequality< f0(z) > 0, z ∈ E.Hence they are univalent for all real numbersαandβsatisfyingα ≤ β < 1and the result is sharp in the sense that the constantβ cannot be replaced by any real number less thanα.

The main objective of this paper is to extend the region of variability of the oper- ator

(1−α)f0(z) +α

1 + zf00(z) f0(z)

,

implying univalence of f ∈ A in E, for 0 < α ≤ 2. We prove a subordination theorem and as applications of the main result, we find the sufficient conditions for f ∈ Ato be univalent, starlike andφ-like.

To prove our main results, we need the following lemma due to Miller and Mo- canu.

Lemma 1.1 ([3, p.132, Theorem 3.4 h]). Letqbe univalent inEand letθandφbe analytic in a domainDcontainingq(E), withφ(w)6= 0, whenw∈q(E).

SetQ(z) =zq0(z)φ[q(z)],h(z) =θ[q(z)] +Q(z)and suppose that either (i) his convex, or

(ii) Qis starlike.

In addition, assume that (iii) < zhQ(z)0(z) >0, z∈E.

Ifpis analytic inE, withp(0) =q(0), p(E)⊂Dand

θ[p(z)] +zp0(z)φ[p(z)]≺θ[q(z)] +zq0(z)φ[q(z)], thenp≺qandqis the best dominant.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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2. Main Result

Theorem 2.1. Letα 6= 0 be a complex number. Let q, q(z) 6= 0, be a univalent function inEsuch that

(2.1) <

1 + zq00(z)

q0(z) − zq0(z) q(z)

>max

0,<

α−1 α q(z)

.

Ifp, p(z)6= 0, z ∈E, satisfies the differential subordination

(2.2) (1−α)(p(z)−1) +αzp0(z)

p(z) ≺(1−α)(q(z)−1) +αzq0(z) q(z) , thenp≺qandqis the best dominant.

Proof. Let us define the functionsθandφas follows:

θ(w) = (1−α)(w−1), and

φ(w) = α w.

Obviously, the functionsθandφare analytic in domainD=C\ {0}andφ(w)6= 0 inD.

Now, define the functionsQandhas follows:

Q(z) = zq0(z)φ(q(z)) =αzq0(z) q(z) , and

h(z) = θ(q(z)) +Q(z) = (1−α)(q(z)−1) +αzq0(z) q(z) .

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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Then in view of condition (2.1), we have (1)Qis starlike inEand

(2)<z hQ(z)0(z) >0, z∈E.

Thus conditions (ii) and (iii) of Lemma1.1, are satisfied.

In view of (2.2), we have

θ[p(z)] +zp0(z)φ[p(z)]≺θ[q(z)] +zq0(z)φ[q(z)].

Therefore, the proof, now, follows from Lemma1.1.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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3. Applications to Univalent Functions

On writingp(z) = f0(z)in Theorem2.1, we obtain the following result.

Theorem 3.1. Letα 6= 0 be a complex number. Let q, q(z) 6= 0, be a univalent function in E and satisfy the condition (2.1) of Theorem 2.1. If f ∈ A, f0(z) 6=

0, z∈E, satisfies the differential subordination (1−α)(f0(z)−1) +αzf00(z)

f0(z) ≺(1−α)(q(z)−1) +αzq0(z) q(z) , thenf0(z)≺q(z)andqis the best dominant.

On writingp(z) = zff(z)0(z) in Theorem2.1, we obtain the following result.

Theorem 3.2. Letα 6= 0 be a complex number. Let q, q(z) 6= 0, be a univalent function in E and satisfy the condition (2.1) of Theorem 2.1. If f ∈ A, zff(z)0(z) 6=

0, z∈E, satisfies the differential subordination (1−2α)zf0(z)

f(z) +α

1 + zf00(z) f0(z)

≺(1−α)q(z) +αzq0(z) q(z) ,

then zff(z)0(z) ≺q(z)andqis the best dominant.

By takingp(z) = φ(fzf0(z))(z) in Theorem2.1, we obtain the following result.

Theorem 3.3. Letα 6= 0 be a complex number. Let q, q(z) 6= 0, be a univalent function inE and satisfy the condition (2.1) of Theorem 2.1. If f ∈ A, φ(f(z))zf0(z) 6=

0, z∈E, satisfies the differential subordination (1−α) zf0(z)

φ(f(z)) +α

1 + zf00(z)

f0(z) − z[φ(f(z))]0 φ(f(z))

≺(1−α)q(z) +αzq0(z) q(z) ,

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whereφis analytic in a domain containingf(E),φ(0) = 0, φ0(0) = 1andφ(w)6= 0 forw∈f(E)\ {0}, then φ(f(z))zf0(z) ≺q(z)andqis the best dominant.

Remark 1. When we select the dominantq(z) = 1+z1−z, z∈E,then

Q(z) = αzq0(z)

q(z) = 2αz 1−z2,

and zQ0(z)

Q(z) = 1 +z2 1−z2. Therefore, we have

<zQ0(z)

Q(z) >0, z ∈E, and henceQis starlike. We also have

1 + zq00(z)

q0(z) − zq0(z)

q(z) + 1−α

α q(z) = 1 +z2

1−z2 + 1−α α

1 +z 1−z. Thus, for any real number0< α≤2,we obtain

<

1 + zq00(z)

q0(z) − zq0(z)

q(z) +1−α α q(z)

>0, z ∈E.

Therefore,q(z) = 1+z1−z, z ∈E,satisfies the conditions of Theorem3.1, Theorem 3.2and Theorem3.3.

Moreover,

(1−α)(q(z)−1) +αzq0(z)

q(z) = 2(1−α) z

1−z + 2α z

1−z2 =F(z).

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For0 < α ≤2,we see thatF is the conformal mapping of the unit diskEwith F(0) = 0and

F(E) = C\n

w∈C:<w=α−1, |=w| ≥p

α(2−α)o .

In view of the above remark, on writingq(z) = 1+z1−z in Theorem3.1, we have the following result.

Corollary 3.4. Iff ∈ A,f0(z)6= 0, z∈E, satisfies the differential subordination

(1−α)f0(z) +α

1 + zf00(z) f0(z)

≺1 + 2(1−α) z

1−z + 2α z 1−z2,

where 0 < α ≤ 2 is a real number, then < f0(z) > 0, z ∈ E. Therefore, f is close-to-convex and hencef is univalent inE.

In view of Remark1and Corollary3.4, we obtain the following result.

Corollary 3.5. Let0 < α ≤ 2 be a real number. Suppose thatf ∈ A, f0(z) 6=

0, z∈E, satisfies the condition

(1−α)f0(z) +α

1 + zf00(z) f0(z)

≺F(z).

Thenf is close-to-convex and hence univalent inE,whereF is the conformal map- ping of the unit diskEwithF(0) = 1and

F(E) =C\n

w∈C:<w=α, |=w| ≥p

α(2−α)o .

From Corollary 3.4, we obtain the following result of Singh, Gupta and Singh [7].

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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Corollary 3.6. Let0 < α < 1 be a real number. If f ∈ A, f0(z) 6= 0, z ∈ E, satisfies the differential inequality

<

(1−α)f0(z) +α

1 + zf00(z) f0(z)

> α,

then<f0(z) > 0, z ∈ E.Therefore,f is close-to-convex and hencef is univalent inE.

From Corollary3.4, we obtain the following result.

Corollary 3.7. Let 1 < α ≤ 2,be a real number. If f ∈ A, f0(z) 6= 0, z ∈ E, satisfies the differential inequality

<

(1−α)f0(z) +α

1 + zf00(z) f0(z)

< α,

then<f0(z) > 0, z ∈ E.Therefore,f is close-to-convex and hencef is univalent inE.

When we selectq(z) = 1+z1−z in Theorem3.2, we obtain the following result.

Corollary 3.8. Iff ∈ A, zff(z)0(z) 6= 0, z ∈E, satisfies the differential subordination

(1−2α)zf0(z) f(z) +α

1 + zf00(z) f0(z)

≺(1−α)1 +z

1−z + 2α z

1−z2 =F1(z), where0< α≤2is a real number, thenf ∈ S.

In view of Corollary3.8, we have the following result.

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Region of Variability of a Subclass of Univalent Functions Sukhwinder Singh, Sushma Gupta

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Corollary 3.9. Let0 < α ≤ 2be a real number. Suppose that f ∈ A, zff(z)0(z) 6=

0, z∈E, satisfies the condition

(1−2α)zf0(z) f(z) +α

1 + zf00(z) f0(z)

≺F1(z).

Thenf ∈ S, whereF1 is the conformal mapping of the unit disk EwithF1(0) = 1−αand

F1(E) = C\n

w∈C:<w= 0, |=w| ≥p

α(2−α)o .

In view of Corollary3.8, we have the following result.

Corollary 3.10. Let 0 < α < 1 be a real number. Iff ∈ A, zff(z)0(z) 6= 0, z ∈ E, satisfies the differential inequality

<

(1−2α)zf0(z) f(z) +α

1 + zf00(z) f0(z)

>0,

thenf ∈ S.

In view of Corollary3.8, we also have the following result.

Corollary 3.11. Let1 < α ≤ 2,be a real number. If f ∈ A, zff(z)0(z) 6= 0, z ∈ E, satisfies the differential inequality

<

(1−2α)zf0(z) f(z) +α

1 + zf00(z) f0(z)

<0, thenf ∈ S.

When we selectq(z) = 1+z1−z in Theorem3.3, we obtain the following result.

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Corollary 3.12. Let0 < α ≤ 2be a real number. Let f ∈ A, φ(f(z))zf0(z) 6= 0, z ∈ E, satisfy the differential subordination

(1−α) zf0(z) φ(f(z))+α

1 + zf00(z)

f0(z) − z[φ(f(z))]0 φ(f(z))

≺(1−α)1 +z

1−z+2α z

1−z2 =F1(z).

Then φ(fzf0(z))(z)1+z1−z, where φ is analytic in a domain containing f(E), φ(0) = 0, φ0(0) = 1andφ(w)6= 0forw∈f(E)\ {0}.

In view of Corollary3.12, we obtain the following result.

Corollary 3.13. Let0 < α ≤ 2be a real number. Let f ∈ A, φ(f(z))zf0(z) 6= 0, z ∈ E, satisfy the condition

(1−α) zf0(z) φ(f(z)) +α

1 + zf00(z)

f0(z) −z[φ(f(z))]0 φ(f(z))

≺F1(z).

Then f is φ-like in E, where φ is analytic in a domain containing f(E), φ(0) = 0, φ0(0) = 1andφ(w)6= 0forw∈f(E)\ {0}andF1 is the conformal mapping of the unit diskEwithF1(0) = 1−αand

F1(E) = C\n

w∈C:<w= 0, |=w| ≥p

α(2−α)o .

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References

[1] H.S. AL-AMIRI AND M.O. READE, On a linear combination of some ex- pressions in the theory of univalent functions, Monatshefto für Mathematik, 80 (1975), 257–264.

[2] L. BRICKMAN,φ-like analytic functions. I, Bull. Amer. Math. Soc., 79 (1973), 555–558.

[3] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathe- matics, (No. 225), Marcel Dekker, New York and Basel, 2000.

[4] K. NOSHIRO, On the theory of schlicht functions, J. Fac. Sci., Hokkaido Univ., 2 (1934-35), 129–155.

[5] St. RUSCHEWEYH, A subordination theorem for φ-like functions, J. London Math. Soc., 2(13) (1976), 275–280.

[6] S. SINGH, S. GUPTA AND S. SINGH, On a problem of univalence of func- tions satisfying a differential inequality, Mathematical Inequalities and Applica- tions, 10(1) (2007), 95–98.

[7] V. SINGH, S. SINGH AND S. GUPTA, A problem in the theory of univalent functions, Integral Transforms and Special Functions, 16(2) (2005), 179–186.

[8] S.E. WARCHAWSKI, On the higher derivatives at the boundary in conformal mappings, Trans. Amer. Math. Soc., 38 (1935), 310–340.

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