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volume 5, issue 2, article 24, 2004.

Received 30 January, 2004;

accepted 02 February, 2004.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

NEIGHBORHOODS OF CERTAIN CLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDER

G. MURUGUSUNDARAMOORTHY AND H. M. SRIVASTAVA

Department of Mathematics

Vellore Institute of Technology (Deemed University) Vellore 632014, Tamil Nadu

India

EMail:gmsmoorthy@rediffmail.com Department of Mathematics and Statistics University of Victoria

Victoria, British Columbia V8W 3P4 Canada

EMail:harimsri@math.uvic.ca

c

2000Victoria University ISSN (electronic): 1443-5756 024-04

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Neighborhoods of Certain Classes of Analytic Functions

of Complex Order

G. Murugusundaramoorthy and H.M. Srivastava

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J. Ineq. Pure and Appl. Math. 5(2) Art. 24, 2004

Abstract

By making use of the familiar concept of neighborhoods of analytic functions, the authors prove several inclusion relations associated with the(n, δ)-neighborhoods of certain subclasses of analytic functions of complex order, which are introduced here by means of the Ruscheweyh derivatives. Special cases of some of these inclusion relations are shown to yield known results.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Analytic functions, Ruscheweyh derivatives, Starlike functions, Convex functions,δ-neighborhood, Inclusion relations.

1. Introduction and Definitions

LetA(n)denote the class of functionsf of the form:

(1.1) f(z) = z−

∞

X

k=n+1

a_kz^k (a_k ≥0;k ∈N\ {1};n ∈N:={1,2,3, . . .}),

which are analytic in the open unit disk

U:={z:z ∈Cand |z|<1}.

Following the works of Goodman [9] and Ruscheweyh [14], we define the (n, δ)-neighborhood of a functionf ∈ A(n)by (see also [2], [3], [4], and [16]) (1.2) N_n,δ(f) :=

(

g ∈ A(n) :g(z) = z−

∞

X

k=n+1

b_kz^k

and

∞

X

k=n+1

k|a_k−b_k| ≤δ )

.

In particular, for the identity function

(1.3) e(z) =z,

we immediately have (1.4) N_n,δ(e) :=

(

g∈ A(n) :g(z) =z−

∞

X

k=n+1

b_kz^k and

∞

X

k=n+1

k|b_k| ≤δ )

.

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The above concept of (n, δ)-neighborhoods was extended and applied re- cently to families of analytically multivalent functions by Altinta¸s et al. [6]

and to families of meromorphically multivalent functions by Liu and Srivastava ([10] and [11]). The main object of the present paper is to investigate the(n, δ)- neighborhoods of several subclasses of the class A(n) of normalized analytic functions in U with negative and missing coefficients, which are introduced below by making use of the Ruscheweyh derivatives.

First of all, we say that a function f ∈ A(n)is starlike of complex order γ (γ ∈C\ {0}), that is,f ∈ S_n^∗(γ), if it also satisfies the following inequality:

(1.5) R

1 + 1

γ

zf⁰(z)

f(z) −1 >0 (z ∈U; γ ∈C\ {0}).

Furthermore, a function f ∈ A(n) is said to be convex of complex order γ (γ ∈C\ {0}),that is,f ∈ C_n(γ), if it also satisfies the following inequality:

(1.6) R

1 + 1

γ

zf⁰⁰(z) f⁰(z)

>0 (z ∈U; γ ∈C\ {0}).

The classesS_n^∗(γ)andCn(γ)stem essentially from the classes of starlike and convex functions of complex order, which were considered earlier by Nasr and Aouf [12] and Wiatrowski [18], respectively (see also [5] and [7]).

Next, for the functionsf_j (j = 1,2)given by

(1.7) f_j(z) = z+

∞

X

k=2

a_k,j z^k (j = 1,2),

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letf₁∗f₂ denote the Hadamard product (or convolution) of f₁ andf₂, defined by

(1.8) (f₁∗f₂) (z) := z+

∞

X

k=2

a_k,1a_k,2 z^k=: (f₂∗f₁) (z).

Thus the Ruscheweyh derivative operator D^λ : A −→ Ais defined for A :=

A(1)by

(1.9) D^λf(z) := z

(1−z)^λ+1 ∗f(z) (λ >−1; f ∈ A)

or, equivalently, by

(1.10) D^λf(z) :=z−

∞

X

k=2

λ+k−1 k−1

akz^k (λ >−1; f ∈ A)

for a functionf ∈ Aof the form (1.1). Here, and in what follows, we make use of the following standard notation:

(1.11)

κ k

:= κ(κ−1)· · ·(κ−k+ 1)

k! (κ∈C; k∈N⁰)

for a binomial coefficient. In particular, we have (1.12) Dⁿf(z) = z zⁿ⁻¹f(z)(n)

n! (n∈N0 :=N∪ {0}).

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Finally, in terms of the Ruscheweyh derivative D^λ(λ > −1) defined by (1.9) or (1.10) above, letSn(γ, λ, β)denote the subclass ofA(n)consisting of functionsf which satisfy the following inequality:

1 γ

z D^λf(z)0

D^λf(z) −1

!

< β (1.13)

(z ∈U; γ ∈C\ {0}; λ >−1; 0 < β≤1).

Also let R_n(γ, λ, β;µ) denote the subclass of A(n) consisting of functionsf which satisfy the following inequality:

1 γ

(1−µ)D^λf(z)

z +µ D^λ f(z)0

−1

< β (1.14)

(z ∈U; γ ∈C\ {0}; λ >−1; 0< β ≤1; 0≤µ≤1).

Various further subclasses of the classesS_n(γ, λ, β)andR_n(γ, λ, β;µ)with γ = 1 were studied in many earlier works (cf., e.g., [8] and [17]; see also the references cited in these earlier works). Clearly, in the case of (for example) the classS_n(γ, λ, β), we have

S_n(γ,0,1)⊂ S_n^∗(γ) and S_n(γ,1,1)⊂ C_n(γ) (1.15)

(n∈N; γ ∈C\ {0}).

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2. Inclusion Relations Involving N

_n,δ

(e)

In our investigation of the inclusion relations involvingNn,δ(e), we shall require Lemma1and Lemma2below.

Lemma 1. Let the functionf ∈ A(n)be defined by(1.1). Thenf is in the class S_n(γ, λ, β)if and only if

(2.1)

∞

X

k=n+1

λ+k−1 k−1

(β|γ|+k−1)a_k ≤β|γ|.

Proof. We first suppose that f ∈ S_n(γ, λ, β). Then, by appealing to the condition (1.13), we readily obtain

(2.2) R z D^λ f(z)0

D^λ f(z) −1

!

>−β|γ| (z ∈U)

or, equivalently,

(2.3) R







−

∞

P

k=n+1

λ+k−1 k−1

(k−1)akz^k z−

∞

P

k=n+1

λ+k−1 k−1

a_kz^k







>−β|γ| (z ∈U),

where we have made use of (1.10) and the definition (1.1). We now choose values of z on the real axis and let z → 1− through real values. Then the

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inequality (2.3) immediately yields the desired condition (2.1). Conversely, by applying the hypothesis (2.1) and letting|z|= 1, we find that

z D^λf(z)⁰ D^λf(z) −1

=

∞

P

k=n+1

λ+k−1 k−1

(k−1)a_kz^k z−

∞

P

k=n+1

λ+k−1 k−1

a_kz^k

≤ β|γ|

1−

∞

P

k=n+1

λ+k−1 k−1

ak

1−

∞

P

k=n+1

λ+k−1 k−1

a_k

≤β|γ|.

(2.4)

Hence, by the maximum modulus theorem, we have f ∈ S_n(γ, λ, β), which evidently completes the proof of Lemma1.

Similarly, we can prove the following result.

Lemma 2. Let the function f ∈ A(n)be defined by (1.1). Then f is in the classR(γ, λ, β;µ)if and only if

(2.5)

∞

X

k=n+1

λ+k−1 k−1

[µ(k−1) + 1]a_k ≤β|γ|.

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Remark 1. A special case of Lemma1when

n = 1, γ = 1, and β= 1−α (0≤α <1) was given earlier by Ahuja [1]. Furthermore, if in Lemma1with

n = 1, γ = 1, and β= 1−α (0≤α <1),

we setλ= 0andλ= 1,we shall obtain the familiar results of Silverman [15].

Our first inclusion relation involvingN_n,δ(e)is given by Theorem1below.

Theorem 1. If

(2.6) δ := (n+ 1)β|γ|

(β|γ|+n)

λ+n n

(|γ|<1),

then

(2.7) S_n(γ, λ, β)⊂N_n,δ(e).

Proof. For a functionf ∈ S_n(γ, λ, β)of the form (1.1), Lemma1immediately yields

(β|γ|+n)

λ+n n

^∞ X

k=n+1

a_k ≤β|γ|,

so that (2.8)

∞

X

k=n+1

a_k≤ β|γ|

(β|γ|+n)

λ+n n

.

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On the other hand, we also find from (2.1) and (2.8) that λ+n

n

^∞ X

k=n+1

ka_k ≤β|γ|+ (1−β|γ|)

λ+n n

^∞ X

k=n+1

a_k

≤β|γ|+ (1−β|γ|)

λ+n n

β|γ|

(β|γ|+n)

λ+n n

≤ (n+ 1)β|γ|

β|γ|+n (|γ|<1), that is,

(2.9)

∞

X

k=n+1

ka_k≤ (n+ 1)β|γ|

(β|γ|+n)

λ+n n

:=δ,

which, in view of the definition (1.4), proves Theorem1.

By similarly applying Lemma2instead of Lemma1, we now prove Theorem 2below.

Theorem 2. If

(2.10) δ:= (n+ 1)β|γ|

(µn+ 1)

λ+n n

,

then

(2.11) Rn(γ, λ, β;µ)⊂Nn,δ(e).

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Proof. Suppose that a functionf ∈ R(γ, λ, β;µ)is of the form (1.1). Then we find from the assertion (2.5) of Lemma2that

λ+n n

(µn+ 1)

∞

X

k=n+1

ak≤β|γ|,

which yields the following coefficient inequality:

(2.12)

∞

X

k=n+1

a_k ≤ β|γ|

(µn+ 1)

λ+n n

.

Making use of (2.5) in conjunction with (2.12), we also have µ

λ+n n

^∞ X

k=n+1

ka_k≤β|γ|+ (µ−1)

λ+n n

^∞ X

k=n+1

a_k

≤β|γ|+ (µ−1)

λ+n n

β|γ|

(µn+ 1)

λ+n n

, (2.13)

that is,

∞

X

k=n+1

kak≤ (n+ 1)β|γ|

(µn+ 1)

λ+n n

=:δ,

which, in light of the definition (1.4), completes the proof of Theorem2.

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Remark 2. By suitably specializing the various parameters involved in Theo- rem 1and Theorem2, we can derive the corresponding inclusion relations for many relatively more familiar function classes (see also Equation (1.15) and Remark1above).

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3. Neighborhoods for the Classes S

_n^(α)

(γ, λ, β) and R

^(α)n

(γ, λ, β; µ)

In this section we determine the neighborhood for each of the classes S_n^(α)(γ, λ, β) and R^(α)_n (γ, λ, β;µ),

which we define as follows. A function f ∈ A(n) is said to be in the class Sn^(α)(γ, λ, β)if there exists a functiong ∈ S_n(γ, λ, β)such that

(3.1)

f(z) g(z) −1

<1−α (z ∈U; 0≤α <1).

Analogously, a functionf ∈ A(n) is said to be in the classR^(α)n (γ, λ, β;µ)if there exists a function g ∈ R_n(γ, λ, β;µ) such that the inequality (3.1) holds true.

Theorem 3. Ifg ∈ S_n(γ, λ, β)and

(3.2) α= 1−

(β|γ|+n)δ

λ+n n

(n+ 1)

(β|γ|+n)

λ+n n

−β|γ|

,

then

(3.3) Nn,δ(g)⊂ S_n^(α)(γ, λ, β).

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Proof. Suppose thatf ∈N_n,δ(g). We then find from the definition (1.2) that (3.4)

∞

X

k=n+1

k|a_k−b_k| ≤δ,

which readily implies the coefficient inequality:

(3.5)

∞

X

k=n+1

|a_k−b_k| ≤ δ

n+ 1 (n∈N).

Next, sinceg ∈ S_n(γ, λ, β),we have [cf. Equation (2.8)]

(3.6)

∞

X

k=n+1

b_k≤ β|γ|

(β|γ|+n)

λ+n n

,

so that

f(z) g(z) −1

<

∞

P

k=n+1

|a_k−b_k| 1−

∞

P

k=n+1

b_k

≤ δ n+ 1 ·

(β|γ|+n)

λ+n n

(β|γ|+n)

λ+n n

−β|γ|

= 1−α, (3.7)

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provided thatαis given precisely by (3.2). Thus, by definition,f ∈ Sn^(α)(γ, λ, β) forαgiven by (3.2). This evidently completes our proof of Theorem3.

Our proof of Theorem4below is much akin to that of Theorem3.

Theorem 4. Ifg ∈ R_n(γ, λ, β;µ)and

(3.8) α = 1−

(µn+ 1)δ

λ+n n

(n+ 1)

(µn+ 1)

λ+n n

−β|γ|

,

then

(3.9) N_n,δ(g)⊂ R^(α)_n (γ, λ, β;µ).

Remark 3. Just as we indicated already in Section1and Remark 2, Theorem 3and Theorem4can readily be specialized to deduce the corresponding neigh- borhood results for many simpler function classes.

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References

[1] O.P. AHUJA, Hadamard products of analytic functions defined by Ruscheweyh derivatives, in Current Topics in Analytic Function Theory, H.M. Srivastava and S. Owa (Eds.), pp. 13–28, World Scientific Publish- ing Company, Singapore, New Jersey, London and Hong Kong, 1992.

[2] O.P. AHUJAANDM. NUNOKAWA, Neighborhoods of analytic functions defined by Ruscheweyh derivatives, Math. Japon., 51 (2003), 487–492.

[3] O. ALTINTA ¸SANDS. OWA, Neighborhoods of certain analytic functions with negative coefficients, Internat. J. Math. and Math. Sci., 19 (1996), 797–800.

[4] O. ALTINTA ¸S, Ö. ÖZKANANDH.M. SRIVASTAVA, Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett., 13(3) (2000), 63–67.

[5] O. ALTINTA ¸S, Ö. ÖZKAN AND H.M. SRIVASTAVA, Majorization by starlike functions of complex order, Complex Variables Theory Appl., 46 (2001), 207–218.

[6] O. ALTINTA ¸S, Ö. ÖZKANANDH.M. SRIVASTAVA, Neighborhoods of a certain family of multivalent functions with negative coefficients, Com- put. Math. Appl., 47 (2004) (to appear).

[7] O. ALTINTA ¸S AND H.M. SRIVASTAVA, Some majorization problems associated withp-valently starlike and convex functions of complex order, East Asian Math. J., 17 (2001), 175–183.

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[8] P.L. DUREN, Univalent Functions, A Series of Comprehensive Studies in Mathematics, Vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1983.

[9] A.W. GOODMAN, Univalent functions and nonanalytic curves, Proc.

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[11] J.-L. LIUANDH.M. SRIVASTAVA, Subclasses of meromorphically mul- tivalent functions associated with a certain linear operator, Math. Comput.

Modelling, 39 (2004), 35–44.

[12] M.A. NASR AND M.K. AOUF, Starlike function of complex order, J.

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[14] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer.

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[15] H. SILVERMAN, Univalent functions with negative coefficients, Proc.

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[16] H. SILVERMAN, Neighborhoods of classes of analytic functions, Far East J. Math. Sci., 3 (1995), 165–169.

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[17] H.M. SRIVASTAVA AND S. OWA (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.

[18] P. WIATROWSKI, On the coefficients of some family of holomorphic functions, Zeszyty Nauk. Uniw. Łódz Nauk. Mat.-Przyrod. (2), 39 (1970), 75–85.