ARGUMENT ESTIMATES FOR CERTAIN ANALYTIC FUNCTIONS ASSOCIATED WITH THE CONVOLUTION STRUCTURE

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Argument Estimates S.P. Goyal, Pranay Goswami

and N. E. Cho vol. 10, iss. 1, art. 20, 2009

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ARGUMENT ESTIMATES FOR CERTAIN ANALYTIC FUNCTIONS ASSOCIATED WITH THE

CONVOLUTION STRUCTURE

S.P. GOYAL, PRANAY GOSWAMI N. E. CHO

Department of Mathematics Department of Applied Mathematics

University of Rajasthan Pukyong National University

Jaipur-302055, India Pusan 608-737, Korea

EMail:somprg@gmail.com,pranaygoswami83@gmail.com EMail:necho@pknu.ac.kr

Received: 15 March, 2008

Accepted: 10 January, 2009 Communicated by: H.M. Srivastava

2000 AMS Sub. Class.: Primary 26A33; Secondary 30C45.

Key words: Argument estimate; Subordination; Univalent function; Hadamard product(or convolution); Dziok-Srivastava operator.

Abstract: The purpose of the present paper is to investigate some argument properties for certain analytic functions in the open unit disk associated with the convolution structure. Some interesting applications are also considered as special cases of main results presented here.

Acknowledgements: The second author is thankful to CSIR, India, for providing Junior Research Fel- lowship under research scheme no. 09/135(0434)/2006-EMR-1.

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1. Introduction

LetAdenote the class of functions of the form

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ (n∈N:={1,2,3, . . .}),

which are analytic in the open unit diskU:={z :|z|<1}.

Iff ∈ Ais given by (1.1) andg ∈ Ais given by g(z) =z+

∞

X

n=2

b_nzⁿ,

then the Hadamard product (or convolution)f ∗goff andg is defined by

(1.2) (f ∗g)(z) := z+

∞

X

n=2

a_nb_nzⁿ=: (g∗f)(z).

We observe that several known operators are deducible from the convolution. That is, for various choices ofg in (1.2), we obtain some interesting operators studied by many authors. For example, for functionsf ∈ Aand the function defined by

g(z) = z+

∞

X

n=2

(α₁)_n−1· · ·(α_q)_n−1

(β₁)_n−1· · ·(β_s)_n−1(n−1)! zⁿ (1.3)

(αi ∈C, βj ∈C\Z₀⁻;Z₀⁻={0,−1,−2, . . .}; i= 1, . . . , q; j = 1, . . . , s;

q≤s+ 1; q, s∈N₀ =N ∪ {0};z ∈U),

the convolution (1.2) with the functiong defined by (1.3) gives the operator studied by Dziok and Srivastava ([5], see also [4,6]):

(1.4) (g∗f)(z) :=H(α1, . . . , αq;β1, . . . , βs)f(z).

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We note that the linear operatorH(α₁, . . . , α_q;β₁, . . . , β_s)includes various other linear operators which were introduced and studied various researchers in the litera- ture.

Next, if we define the functiongby

(1.5) g(z) =z+

∞

X

n=2

n+λ 1 +λ

k

zⁿ (λ≥0; k ∈Z),

then for functionsf ∈ A, the convolution (1.2) with the functiong defined by (1.5) reduces to the multiplier transformation studied by Cho and Srivastava [2]:

(1.6) (g∗f)(z) :=I_λ^kf(z).

For arbitrary fixed real numbers A and B (−1 ≤ B < A ≤ 1), we denote by P(A, B)the class of functions of the form

q(z) = 1 +c₁z+· · ·,

which are analytic in the unit diskUand satisfies the condition

(1.7) q(z)≺ 1 +Az

1 +Bz (z ∈U),

where the symbol≺stands for usual subordination. We note that the classP(A, B) was introduced and studied by Janowski [9].

We also observe from (1.7) (see, also [11]) that a functionq(z)∈P(A, B)if and only if

(1.8)

q(z)− 1−AB 1−B²

< A−B

1−B² (B 6=−1; z ∈U)

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and

(1.9) Re{q(z)}> 1−A

2 (B =−1; z ∈U).

In the present paper, we obtain some argument properties for certain analytic functions in A associated with the convolution structure by using the techniques involving the principle of differential subordination. Relevant connections of the results, which are presented in this paper, with various known operators are also considered.

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2. Main Results

Theorem 2.1. Letf, g ∈Aandβ ≥0, 0< η ≤1. Suppose also that (2.1) z(g∗h)⁰(z)

(g∗h)(z) ≺ 1 +Az

1 +Bz (h∈ A; −1≤B < A≤1; z ∈U).

If

arg

β(g∗f)⁰(z)

(g∗h)⁰(z) + (1−β)(g∗f)(z) (g∗h)(z)

< π

2η (z ∈U),

then

arg

(g∗f)(z) (g∗h)(z)

< π

2α (z ∈U), whereα(0< α≤1)is the solution of the equation given by

(2.2) η=







α+_π² tan⁻¹ ^βα^sin

π

2(1−t(A,B))

1+A

1+B+βαcos^π₂(1−t(A,B)) forB 6=−1,

α forB =−1,

and

(2.3) t(A, B) = 2

π sin⁻¹

A−B 1−AB

.

Proof. Let

p(z) = (g∗f)(z)

(g∗h)(z) and q(z) = z(g∗h)⁰(z) (g∗h)(z) . Then by a simple calculation, we have

β(g∗f)⁰(z)

(g∗h)⁰(z) + (1−β)(g∗f)(z)

(g∗h)(z) =p(z) + βzp⁰(z) q(z) .

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While, from the assumption (2.1) with (1.8) and (1.9), we obtain q(z) =ρe^πθ² ⁱ,

where







1−A

1−B < ρ < ^1+A_1+B

−t(A, B)< θ < t(A, B) forB 6=−1, whent(A, B)is given by (2.3) and







1−A

2 < ρ <∞

−1< θ <1 forB =−1.

The remaining part of the proof of the Theorem2.1follows by known results due to Miller and Mocanu [9] and Nunokawa [10] and applying a method similar to that of Cho et al. [3, Proof of Theorem 2.3], so we omit the details.

In particular, if we putg(z) = z/(1−z)in Theorem2.1, we have the following result.

Corollary 2.2. Letf ∈ Aandβ >0, 0< η≤1. If

arg

βf⁰(z) + (1−β)f(z) z

< π 2η,

then

arg

f(z) z

< π 2α,

whereα(0< α <1)is the solution of the equation given by

(2.4) η=α+ 2

πtan⁻¹(αβ).

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Theorem 2.3. Letf, g, h ∈ Aandµ >0,0< η <1. If (2.5)

arg

(g∗h)(z) (g∗f)(z)

µ

1 + z(g∗f)⁰(z)

(g∗f)(z) − z(g∗h)⁰(z) (g∗h)(z)

< π

2η (z ∈U)

then

arg

(g∗f)(z) (g∗h)(z)

µ

< π

2α (z ∈U), whereα(0< α <1)is the solution of the equation given by

(2.6) η =−α+ 2

πtan⁻¹ α µ. Proof. Let

(2.7) p(z) =

(g∗f)(z) (g∗h)(z)

µ

(µ >0;z ∈U).

By differentiating both sides of (2.7) logarithmically and simplifying, we get (g∗h)(z)

(g∗f)(z) µ

1 + z(g∗f)⁰(z)

(g ∗f)(z) − z(g∗h)⁰(z) (g∗h)(z)

= 1

p(z)

1 + zp⁰(z) µp(z)

. Now by using a lemma due to Nunokawa [10] and a method similar to the proof of Theorem2.1, we get Theorem2.3.

Setting(g∗h)(z) =zandg(z) =z/(1−z)in Theorem2.3, we obtain Corollary 2.4below which is comparable to the result studied by Lashin [8].

Corollary 2.4. Letf, g ∈ Aand0< µ, η <1. If

arg

"

z (g∗f)(z)

µ+1

(g∗f)⁰(z)

#

< π

2η (z ∈U),

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then

arg

(g∗f)(z) z

µ

< π

2α (z ∈U), whereα(0< α <1)is the solution of the equation given by (2.6).

Theorem 2.5. Letf, g ∈ Aandβ >0, 0< η ≤1. If

(2.8)

arg

z(g∗f)⁰(z)

ϕ[(g ∗f)(z)] 1 +βz(g∗f)⁰⁰(z)

(g∗f)⁰(z) −βzϕ⁰[(g∗f)(z)]

ϕ[(g∗f)(z)]

< π 2η, where ϕ[w] is analytic in (g ∗f)(U), ϕ[0] = ϕ⁰[0] −1 = 0 and ϕ[w] 6= 0 in (g∗f)(U)\{0}, then

arg

z(g∗f)⁰(z) ϕ(g∗f)(z)

< π 2α,

whereα(0< α <1)is the solution of the equation given by (2.4).

Proof. Our proof of Theorem 2.5 is much akin to that of Theorem 2.3. Indeed in place (2.7) we definep(z)by

(2.9) p(z) =

(g∗f)(z) (g∗h)(z)

µ

(µ >0;z ∈U).

We choose to skip the detailed involved.

By settingϕ[(g∗f)(z)] = (g∗f)(z)andg(z) = z/(1−z), we have the following result.

Corollary 2.6. Letf ∈ Aandβ >0, 0< η≤1. If

arg

zf⁰(z)

f(z) 1 +βzf⁰⁰(z)

f⁰(z) −βzf⁰(z) f(z)

< π

2η (z ∈U)

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then

arg

zf⁰(z) f(z)

< π

2α (z ∈U),

whereα(0< α <1)is the solution of the equation given by (2.4).

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3. Some Remarks and Observations

Using the Hadamard product (or convolution) defined by (1.2) and applying the differential subordination techniques, we obtained some argument properties of nor- malized analytic functions in the open unit diskU. If we replacegin Theorems2.1, 2.3and2.5by the function H(α1, . . . , αq;β1, . . . , βs)defined by (1.4) or the multiplier transformationI_λ^k defined by (1.5), then we have the corresponding results to the Theorems2.1, 2.3and2.5. Moreover, we note that, if we suitably chooseϕ introduced in Theorem2.5(which is called theϕ-like function [1]), then we can obtain various interesting applications.

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References

[1] L. BRICKMAN, Action of a force near a planar surface between two semi- infiniteϕ-like analytic functions, Bull. Amer. Math. Soc., 79 (1973), 555–558.

[2] N.E. CHO ANDH. M. SRIVASTAVA, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Appl. Math. Com- put., 103 (1999), 39–49.

[3] N.E. CHO, J. PATELAND R.M. MOHPATRA, Argument estimates of certain multi-valent functions involving a linear operator, International J. Math. Math.

Sci., 31(11) (2002), 659–673.

[4] J. DZIOK AND H. M. SRIVASTAVA, Classes of analytic functions associ- ated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13.

[5] J. DZIOK AND H. M. SRIVASTAVA, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeo- metric function, Adv. Stud. Contemp. Math., 2 (2002), 115–125.

[6] J. DZIOKANDH. M. SRIVASTAVA, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14 (2003), 7–18.

[7] W. JANOWSKI, Some extremal problems for certain families of analytic func- tion I, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 17–23.

[8] A. Y. LASHIN, Applications of Nunokawa’s theorem, J. Inequal. Pure Appl.

Math., 5(4) (2004), Art. 111. [ONLINE: http://jipam.vu.edu.au/

article.php?sid=466]

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[9] S.S. MILLERAND P.T. MOCANU, Differential subordinations and inequali- ties in the complex plane, J. Differential Equations, 67 (1987), 199–211.

[10] M. NUNOKAWA, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 234–237.

[11] H. SILVERMANAND E.M. SILIVIA, Subclasses of starlike functions subor- dinate to convex functions, Canad. J. Math., 37 (1985), 48–61.