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volume 7, issue 3, article 103, 2006.

Received 06 October, 2005;

accepted 09 March, 2006.

Communicated by:G. Kohr

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Journal of Inequalities in Pure and Applied Mathematics

ON NEIGHBORHOODS OF A CERTAIN CLASS OF COMPLEX ORDER DEFINED BY RUSCHEWEYH DERIVATIVE OPERATOR

ÖZNUR ÖZKAN AND OSMAN ALTINTA ¸S

Department of Statistics and Computer Sciences Baskent University

Baglıca, TR 06530 Ankara, Turkey.

EMail:oznur@baskent.edu.tr Department of Mathematics Education Baskent University

Baglica, TR 06530 Ankara, Turkey.

EMail:oaltintas@baskent.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 304-05

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On Neighborhoods of a Certain Class of Complex Order Defined

by Ruscheweyh Derivative Operator

Öznur Özkan and Osman Altinta¸s

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J. Ineq. Pure and Appl. Math. 7(3) Art. 103, 2006

Abstract

In this paper, we introduce the subclassR^λ_b(A, B, α, µ)which is defined by con- cept of subordination. According to this, we obtain a necessary and sufficient condition which is equivalent to this class. Further, we apply to theδ−neighborhoods for belonging toR^λ_b(A, B, α, µ)to this condition.

2000 Mathematics Subject Classification:30C45.

Key words: Analytic function, Hadamard product,δ−neighborhood, Subordination, Close-to-convex function.

1. Introduction and Definitions

Let U={z :z ∈Cand |z|<1}andH(U) be the set of all functions analytic inU,and let

A:={f ∈ H(U) :f(0) =f⁰(0)−1 = 0}.

Given two functionsf andg, which are analytic inU. The functionf is said to be subordinate tog, written

f ≺g and f(z)≺g(z) (z ∈U), if there exists a Schwarz functionωanalytic inU, with

ω(0) = 0 and |ω(z)|<1 (z ∈U), and such that

f(z) =g(ω(z)) (z ∈U).

In particular, ifg is univalent in U, thenf ≺ g if and only iff(0) = g(0) andf(U)⊂g(U)in [7].

Next, for the functions f_j (j = 1,2)given by fj(z) =z+

∞

X

k=2

ak,jz^k (j = 1,2).

Let f1∗f2denote the Hadamard product (or convolution) of f1andf2, defined by

(1.1) (f₁ ∗f₂) (z) :=z+

∞

X

k=2

a_k,1a_k,2z^k =: (f₂∗f₁) (z).

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(a)_v denotes the Pochhammer symbol (or the shifted factorial), since (1)_n=n! for n∈N0 :=N∪ {0},

defined (fora, v∈Cand in terms of the Gamma function) by (a)_v := Γ (a+v)

Γ (a) =

( 1; (v = 0, a∈C\ {0}),

a(a+ 1). . .(a+n−1) ; (v =n∈N;a∈C). The earlier investigations by Goodman [1] and Ruscheweyh [9], we define theδ−neighborhood of a functionf ∈ Aby

Nδ(f) :=

(

g ∈ A:f(z) =z+

∞

X

k=2

akz^k ,

g(z) =z+

∞

X

k=2

b_kz^k and

∞

X

k=2

k|a_k−b_k| ≤δ )

so that, obviously, N_δ(e) :=

(

g ∈ A : g(z) =z+

∞

X

k=2

b_kz^k and

∞

X

k=2

k|b_k| ≤δ )

,

wheree(z) :=z.

Ruscheweyh [8] introduced an linear operatorD^λ :A −→ A,defined by the Hadamard product as follows:

D^λf(z) := z

(1−z)^λ+1 ∗f(z) (λ >−1; z ∈U),

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which implies that

Dⁿf(z) = z(zⁿ⁻¹f(z))⁽ⁿ⁾

n! (n∈N0 :=N∪ {0}).

Clearly, we have

D⁰f(z) = f(z), D¹f(z) = zf⁰(z)

and

Dⁿf(z) =

∞

X

k=0

(λ+ 1)_k

(1)_k ak+1z^k+1 =

∞

X

k=0

(λ+ 1)_k

(1)_k z^k+1∗f

! (z),

wheref ∈ A.

Therefore, we can write the following equality, the easily verified result from the above definitions:

(1.2)

(1−µ)D^λf(z)

z +µ D^λf(z)0

∗ 1 (1−z)²

= D^λf(z)⁰

+µz D^λf(z)⁰⁰ , where f ∈ A, λ(λ >−1), µ(µ≥0)and for allz ∈U.

For eachAand B such that −1 ≤ B < A ≤ 1and for all real numbersα such that0≤α <1,we define the function

h(A, B, α;z) := 1 +{(1−α)A+αB}z

1 +Bz (z ∈U).

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Also, let h(α)denote the extremal function of functions with positive real part of orderα(0≤α <1), defined by

h(α) :=h(1,−1, α;z) = 1 + (1−2α)z

1−z (z ∈U).

The class R^b(A, B) is studied by Premabai in [3]. According to this, we introduce the subclassR^λ_b (A, B, α, µ)which is a generalization of this class, as follows:

(1.3) 1 + 1 b

h

D^λf(z)⁰

+µz D^λf(z)⁰⁰

−1 i

≺h(A, B, α;z),

where f ∈ A, b ∈C/{0}, for some real numbersA, B(−1≤B < A≤1), λ (λ >−1), α(0≤α <1), µ(µ≥0)and for allz ∈UwithR_b(A, B, α, µ) :=

R⁰_b(A, B, α, µ)andR^b(A, B) :=R_b(A, B,0,0).

We note that the classR^b(µ) := R_b(1,−1,0, µ)is studied by Altınta¸s and Özkan in [4]. ThereforeC(b) := R_b(1,−1,0,0)is the class of close-to-convex functions of complex order b. C(α) :=R1−α(1,−1,0,0)is the class of close- to-convex functions of orderα(0≤α <1).

Also, let T_b(A, B, α)denote the class of functionsφnormalized by

(1.4) φ(z) :=

1 + ¹_b n

1

(1−z)² −1o

−1+{(1−α)A+αB}e^it 1+Be^it

1− 1+{(1−α)A+αB}e^it 1+Be^it

(t ∈(0,2π)),

where b ∈ C/{0}, for some real numbers A, B (−1≤B < A≤1), for all α (0≤α <1)and for all z ∈ U withT_b(A, B) := T_b(A, B,0)andT (b) :=

T_b(1,−1,0).

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2. The Main Results

A theorem that contains the relationship between the above classes is given as follows:

Theorem 2.1. f ∈ R^λ_b (A, B, α, µ)if and only if

(1−µ)D^λf(z)

z +µ D^λf(z)⁰

∗φ(z)6= 0

for allφ ∈ T_b(A, B, α)and for all f ∈ A.

Proof. Firstly, let

z^λ(f, µ;z) := (1−µ)D^λf(z)

z +µ D^λf(z)⁰ and we suppose that

(2.1) z^λ(f, µ;z)∗φ(z)6= 0

for all f ∈ Aand for allφ ∈ Tb(A, B, α). In view of(1.4),we have z^λ(f, µ;z)∗φ(z)

=

1 + ¹_bn

z^λ(f, µ;z)∗ _(1−z)¹ 2 −1o

− 1+{(1−α)A+αB}e^it 1+Be^it

1− 1+{(1−α)A+αB}e^it 1+Be^it

=

1 + ¹_bn

D^λf(z)0

+µz D^λf(z)00

−1o

− 1+{(1−α)A+αB}e^it 1+Be^it

1− 1+{(1−α)A+αB}e^it 1+Be^it

6= 0.

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From this inequality we find that 1 + 1

b

n D^λf(z)⁰

+µz D^λf(z)⁰⁰

−1o

6=h A, B, α;e^it ,

wheret∈(0,2π).

This means that 1 + ¹_b n

D^λf(z)0

+µz D^λf(z)00

−1o

does not take any value on the image of under h(A, B, α;z)function of the boundary of U. Therefore we note that1+¹_bn

D^λf(z)⁰

+µz D^λf(z)⁰⁰

−1o

takes the value 1forz = 0. Since0 ≤ α < 1and B < A,1is contained by the image under h(A, B, α;z)function ofU.Thus, we can write

1 + 1 b

n D^λf(z)0

+µz D^λf(z)00

−1o

≺h(A, B, α;z).

Hence f ∈ R^λ_b (A, B, α, µ).

Conversely, assume the functionf is in the classR^λ_b (A, B, α, µ).From the definition of subordination, we can write the following inequality:

1 + 1 b

n D^λf(z)⁰

+µz D^λf(z)⁰⁰

−1o

6=h A, B, α;e^it ,

wheret∈(0,2π).From(1.2)we can write 1 + 1

b

z^λ(f, µ;z)∗ 1

(1−z)² −1

6=h A, B, α;e^it

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or equivalently,

z^λ(f, µ;z)∗







1 + ¹_b

1

(1−z)² −1

−h(A, B, α;e^it) 1−h(A, B, α;e^it)





 .

Thus, from the definition of the functionφ,we can write

(1−µ)D^λf(z)

z +µ D^λf(z)0

∗φ(z)6= 0

for allφ ∈ T_b(A, B, α)and for all f ∈ A.

Corollary 2.2. f ∈ R_b(A, B, α, µ) if and only if h

(1−µ)^f^(z)_z +µf⁰(z)i

∗ φ(z)6= 0for allφ∈ T_b(A, B, α)and for allf ∈ A.

Proof. By puttingλ= 0in Theorem2.1.

Corollary 2.3. f ∈ R^b(A, B) if and only if ^f^(z)_z ∗φ(z) 6= 0 for all φ ∈ T_b(A, B)and for allf ∈ A.

Proof. By putting α = 0, µ = 0in Corollary2.2. And, we obtain the result of Theorem 1 in [3].

Corollary 2.4. f ∈ C(b)if and only if ^f^(z)_z ∗φ(z)6= 0for allφ ∈ T (b)and for allf ∈ A.

Proof. By puttingA= 1, B =−1in Corollary2.3.

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Theorem 2.5. Letz(z) = ^f^(z)+z₁₊ for∈Candf ∈ A.If z ∈ R^λ_b (A, B, α, µ) for||< δ^∗, then

(2.2)

(1−µ)D^λf(z)

z +µ D^λf(z)⁰

∗φ(z)

≥δ^∗ (z ∈U),

whereφ∈ T_b(A, B, α)and for allf ∈ A.

Proof. Letφ∈ T_b(A, B, α)andz ∈ R^λ_b (A, B, α, µ).From Theorem2.1, we can write

(1−µ)D^λz(z)

z +µ D^λz(z)0

∗φ(z)6= 0.

Using D^λ(z) =z,we find that the following inequality 1

1 +

(1−µ)D^λf(z)

z +µ D^λf(z)0

∗φ(z) +

6= 0 that is,

(1−µ)D^λf(z)

z +µ D^λf(z)⁰

∗φ(z)6=−

or equivalently(2.2).

Lemma 2.6. Ifφ(z) = 1 +P∞

k=1c_kz^k∈ T_b(A, B, α), then we have

(2.3) |c_k| ≤ (k+ 1) (1 +|B|)

(1−α)|b| |B−A| (k = 1,2,3, . . .).

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Proof. We suppose that φ(z) = 1 +P∞

k=1c_kz^k ∈ T_b(A, B, α). From(1.4), we have

1 +

∞

X

k=1

c_kz^k

= 1 + ¹_b

1 +P∞

k=2kz^k−1−1 − 1+{(1−α)A+αB}e^it 1+Be^it

1− 1+{(1−α)A+αB}e^it 1+Be^it

(t∈(0,2π)).

We write the following equality result which is easily verified result from the above equality:

ck= (k+ 1)

b(1−α).(1 +Be^it) (B−A)e^it.

Taking the modulus of both sides, we obtain inequality(2.3). Theorem 2.7. Ifz ∈ R^λ_b (A, B, α, µ)for||< δ^∗,then

N_δ (f)⊂ R^λ_b (A, B, α, µ),

whereδ:= (1−α)|b||B−A|

(1+λ)(1+µ)(1+|B|)δ^∗.

Proof. Letg ∈ N_δ(f) forδ= (1−α)|b||B−A|

(1+λ)(1+µ)(1+|B|)δ^∗. If we take

z^λ(g, µ;z) := (1−µ)D^λg(z)

z +µ D^λg(z)0

,

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then

z^λ(g, µ;z)∗φ(z)

=

(1−µ)D^λ(f +g−f) (z)

z +µ D^λ(f+g−f) (z)0

∗φ(z)

≥

z^λ(f, µ;z)∗φ(z) −

z^λ(g−f, µ;z)∗φ(z) and using Theorem2.5we can write

(2.4) ≥δ^∗−

∞

X

k=2

Ψ (k) (bk−ak)ck−1z^k−1 ,

where

Ψ (k) = (λ+ 1)_(k−1)

(1)_k (µk−µ+ 1). We know thatΨ (k)is an increasing function ofkand

0<Ψ (2) = (1 +λ) (1 +µ)≤Ψ (k)

µ≥0; k∈N; λ≥ −µk (1 +µk)

.

Sincez ∈ R^λ_b (A, B, α, µ)for||< δ^∗ and using Lemma2.6in(2.4)we have

z^λ(g, µ;z)∗φ(z)

> δ^∗−Ψ (2)|z|

∞

X

k=2

|a_k−b_k| k(1 +|B|) (1−α)|b| |B−A|

> δ^∗−(1 +λ) (1 +µ) (1 +|B|) (1−α)|b| |B−A|

∞

X

k=2

k|a_k−b_k|

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> δ^∗−δ(1 +λ) (1 +µ) (1 +|B|) (1−α)|b| |B−A| >0.

That is, we can write

(1−µ)D^λg(z)

z +µ D^λg(z)0

∗φ(z)6= 0 (z ∈U).

Thus, from Theorem2.1we can find thatg ∈ R^λ_b (A, B, α, µ). Corollary 2.8. Ifz ∈ R_b(A, B, α, µ)for||< δ^∗,then

Nδ (f)⊂ Rb(A, B, α, µ), whereδ:= (1−α)|b||B−A|

(1+µ)(1+|B|)δ^∗.

Proof. By puttingλ= 0in Theorem2.7.

Corollary 2.9. Ifz ∈ R_b(A, B)for||< δ^∗,then Nδ (f)⊂ Rb(A, B) whereδ:= ^|b||B−A|_(1+|B|)δ^∗.

Proof. By puttingα = 0, µ= 0in Corollary2.8. Thus, we obtain the result of Theorem2.7in [3].

Corollary 2.10. Ifz ∈ C(b)for||< δ^∗,then N_δ(f)⊂ C(b), whereδ:=|b|δ^∗.

Proof. By puttingA= 1, B =−1in Corollary2.9.

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References

[1] A.W. GOODMAN, Univalent functions and non analytic curves, Proc.

Amer. Math. Soc., 8 (1957), 598–601.

[2] K.K. DIXIT AND S.K. PAL, On some class of univalent functions related to complex order, Indian J. Pure and Appl. Math., 26(9) (1995), 889–896.

[3] M. PREMABAI, On the neighborhoods of a subclass of univalent functions related to complex order, South. Asian Bull. Math., 26 (2002), 71–75.

[4] O. ALTINTA ¸S AND Ö. ÖZKAN, Starlike, convex and close-to-convex functions of complex order, Hacettepe Bull. of Natur.Sci. and Eng., 28 (1999), 37–46.

[5] O. ALTINTA ¸S, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of a class of analytic functions with negative coefficients, App. Math. Lett., 13(3) (2000), 63–67.

[6] O. ALTINTA ¸S, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of a certain family of multivalent functions with negative coefficients, Comp.

Math. Appl., 47 (2004), 1667–1672.

[7] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Ap- plied Mathematics, Vol. 225, Marcel Dekker, New York, (2000).

[8] St. RUSCHEWEYH, New criteria for univalent fınctions, Proc. Amer. Math.

Soc., 49 (1975), 109–115.

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[9] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer.

Soc., 81 (1981), 521–527.