Several interesting consequences of the main results are also mentioned

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http://jipam.vu.edu.au/

Volume 6, Issue 1, Article 9, 2005

SOME INEQUALITIES EXHIBITING CERTAIN PROPERTIES OF SOME SUBCLASSES OF MULTIVALENTLY ANALYTIC FUNCTIONS

HÜSEYIN IRMAK AND R.K. RAINA DEPARTMENTOFMATHEMATICS

FACULTYOFEDUCATION

BASKENTUNIVERSITY

TR-06530, BAGLICACAMPUS

BAGLICA, ETIMESGUT- ANKARA, TURKEY. [email protected] DEPARTMENTOFMATHEMATICS

COLLEGEOFTECHNOLOGY ANDENGINEERING

M.P. UNIVERSITYOFAGRICULTUREANDTECHNOLOGY

UDAIPUR- 31300, INDIA

[email protected]

Received 01 May, 2004; accepted 27 October, 2004 Communicated by N.K. Govil

ABSTRACT. This paper introduces a new subclass and investigates the sufficiency conditions for a function to belong to this subclass. Certain types of inequalities are also studied exhibiting the well-known geometric properties of multivalently analytic functions in the unit disk. Several interesting consequences of the main results are also mentioned.

Key words and phrases: Open unit disk, analytic, multivalently analytic functions, multivalently starlike, multivalently con- vex, rational and complex inequalities, rational functions with complex variable and Jack’s Lemma.

2000 Mathematics Subject Classification. 30C45, 30C50, 30A10.

1. INTRODUCTION AND DEFINITION

LetT(p)denote the class of functionsf(z)of the form:

(1.1) f(z) = z^p+

∞

X

k=p+1

a_kz^k (p∈ N ={1,2,3, . . .}),

ISSN (electronic): 1443-5756 c

This paper is funded, in part, by TÜBITAK (The Scientific and Technical Research Council of Turkey) and Baþkent University (Ankara, Turkey). The first author would like to acknowledge and express his thanks to Professor (Dr.) Mehmet Haberal, Rector of Ba¸skent University, who generously supports scientific researches in all aspects.

The second author’s work was supported by All India Council of Technical Education (Govt. of India), New Delhi.

088-04

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which are analytic and multivalent in the open disk U = {z : z ∈ C and |z| < 1}. A functionf(z) belonging toT(p)is said to be multivalently starlike order αin U if it satisfies the inequality:

(1.2) <

zf⁰(z) f(z)

> α (z ∈ U; 06α < p; p∈ N),

and, a functionf(z)∈ T(p)is said to be multivalently convex of orderαinU if it satisfies the inequality:

(1.3) <

1 + zf⁰⁰(z) f⁰(z)

> α (z ∈ U; 06α < p; p∈ N).

For the aforementioned definitions, one may refer to [1] (see also [11]). Further, a function f(z)∈ T(p)is said to be in the subclassT SK_λ^δ(p;α)if it satisfies the inequality:

<

(

zf⁰(z) +λz²f⁰(z) (1−λ)f(z) +λzf⁰(z)

δ)

> α, (1.4)

(z ∈ U; δ 6= 0; 06λ61; 06α < p; p∈ N).

Here, and throughout this paper, the value of expressions like zf⁰(z) +λz²f⁰⁰(z)

(1−λ)f(z) +λzf⁰(z) δ

,

is considered to be its principal value. We mention below some of the subclasses of the functions T(p)from the families of functionsT SK^δ_λ(p;α)(defined above). Indeed, we have

T S^δ(p;α)≡ T SK^δ₀(p;α) (δ 6= 0, 06α < p, p∈ N), (1.5)

T K^δ(p;α)≡ T SK^δ₁(p;α) (δ 6= 0, 06α < p, p∈ N), (1.6)

T_λ(p;α)≡ T SK¹_λ(p;α) (06λ61, 06α < p, p∈ N) (see [5]).

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The important subclasses in Geometric Function Theory such as multivalently starlike functions S_p(α) of orderα (0 6 α < p;p ∈ N) inU, multivalently convex functions K_p(α) of order α (0 6 α < p; p ∈ N)in U, multivalently starlike functionsS_p in U, multivalently convex functionsK_pinU, starlike functionsS(α)of orderα(06α <1)inU, convex functionsK(α) of orderα(0 6α < 1)inU, starlike functionsS inU and convex functionsKinU, are seen to be easily identifiable with the aforementioned classes ([1], [5] and [11]).

By introducing a subclass T SK^δ_λ(p;α)of functions f(z) ∈ T(p) satisfying the inequality (1.4), our motive in this paper is to obtain sufficient conditions for a function to belong to the above subclass. The other results investigated include certain inequalities for multivalent functions depicting the properties of starlikeness, close-to-convexity and convexity in the open unit disk. Several corollaries are deduced as worthwhile consequences of our main results.

2. MAINRESULTS

Before stating and proving our main results, we require the following assertion (popularly known as Jack’s Lemma).

Lemma 2.1 ([7]). Let the function w(z)be non-constant and regular in the unit disc U such thatw(0) = 0. If|w(z)|attains its maximum value on the circle|z| = r < 1at the point z₀, then

(2.1) z₀w⁰(z₀) =c w(z₀) (c>1).

We begin now to prove the following:

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Theorem 2.2. Letδ∈R\ {0},06α < p, p∈ N andf(z)∈ T(p). If a functionF(z)defined by

(2.2) F(z) = (1−λ)f(z) +λzf⁰(z) (06λ61), satisfies the inequality:

(2.3) <





 1 +z

F⁰⁰(z)

F⁰(z) − ^F_F⁰_(z)^(z) 1−p^δ

zF⁰(z) F(z)

−δ













< ¹_δ whenδ >0

> ¹_δ whenδ <0







(z ∈ U),

thenf(z)∈ T SK^δ_λ(p;β), where β=p^δ−(p−α)^δ.

Proof. Letf(z)∈ T(p)andF(z)be defined by (2.2) . From (1.1) and (2.2), we have zF⁰(z)

F(z) = zf⁰(z) +λ z²f⁰⁰(z) (1−λ)f(z) +λ zf⁰(z) (2.4)

=

p+P∞ k=p+1

k[1+λ(k−1)]

1+λ(p−1) a_kz^k−p 1 +P∞

k=p+1

1+λ(k−1)

1+λ(p−1)a_kz^k−p . (z ∈ U; 06λ61; p∈ N) Now, define a functionw(z)by

(2.5)

zF⁰(z) F(z)

δ

−p^δ = (p−α)^δw(z), (z ∈ U; δ6= 0; 06α < p; p∈ N),

then the functionw(z)is analytic inU andw(0) = 0.Differentiation of (2.5) gives

(2.6) 1 + zF⁰⁰(z)

F⁰(z) − zF⁰(z) F(z) =

(p−α)^δ p^δ+ (p−α)^δw(z)

z w⁰(z)

δ .

Hence, (2.5) and (2.6) yields (2.7)

1 +z

F⁰⁰(z)

F⁰(z) −^F_F⁰_(z)^(z) 1−p^δ_zF₀_(z)

F(z)

^−δ = zw⁰(z) δw(z).

We claim that|w(z)| < 1inU. For otherwise (by Jack’s Lemma), there exists a pointz₀ ∈ U such that

z0w⁰(z0) = c w(z0), where |w(z0)|= 1 (c>1).

Therefore, (2.7) yields (2.8) <







1 +z

F⁰⁰(z)

F⁰(z) −^F_F⁰_(z)^(z) 1−p^δ

zF⁰(z) F(z)

−δ

z=z0







= 1 δ<

z₀w⁰(z₀) w(z₀)

= c δ







> ¹_δ whenδ > 0 6 ¹_δ whenδ < 0, which contradicts our assumption (2.3) . Therefore,|w(z)|<1holds true for allz ∈ U, and we conclude from (2.5) that

(2.9)

zF⁰(z) F(z)

δ

−p^δ

= (p−α)^δ|w(z)|<(p−α)^δ,

which evidently implies that

(2.10) <

(

zF⁰(z) F(z)

δ)

> p^δ−(p−α)^δ,

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and hencef(z)∈ T SK^δ_λ(p;α).

Theorem 2.3. Let δ ∈ R\ {0}; 0 6 α < p; n, m, p ∈ N; q = n −m; f(z) ∈ T(n) andg(z)∈ T(m).Iff(z)satisfies the inequality:

(2.11) <

zf⁰(z) f(z)







< q +α+ _2δ¹ when δ >0 andg(z)∈ S_m(α)

> q +α+ _2δ¹ when δ <0 andg(z)∈ S/ _m(α), then

(2.12) <

(

z^−qf(z) g(z)

δ)

>0,

where the value of

z^{−q f}_g(z)^(z)δ

is taken to be its principle value.

Proof. Letf(z)∈ T(n)andg(z)∈ T(m)withn−m∈ N. Since f(z)

g(z) =z^q+c₁z^q+1+c₂z^q+2+· · · ∈ T(q) (q=n−m ∈ N), we definew(z)by

(2.13)

z^−qf(z) g(z)

δ

= 1 +w(z) (z ∈ U;δ6= 0).

It is clear that the function w(z) is an analytic function in U and w(0) = 0. Differentiating (2.13), we have

(2.14) zf⁰(z)

f(z) =q+ zw⁰(z)

δ(1 +w(z)) +zg⁰(z) g(z) .

If we suppose that there exists a point z₀ ∈ U such thatz₀w⁰(z₀) = c w(z₀)where|w(z₀)| = 1 (c>1), i.e. w(z₀) =e^iθ (θ ∈[0,2π)− {π}),then

<

z₀f⁰(z₀) f(z₀)

=q+ 1 δ<

z₀w⁰(z₀)

1 +w(z₀)+ δ z₀g⁰(z) g(z₀)

=q+ 1 δ<

ce^iθ 1 +e^iθ

+<

z0g⁰(z) g(z₀)

. (2.15)

From (2.15) it follows that

(2.16) <

z₀f⁰(z₀) f(z₀)

>q+α+ 1

2δ (δ >0), provided that

<

z₀g⁰(z₀) g(z₀)

> α, and

(2.17) <

z₀f⁰(z₀) f(z₀)

6q+α+ 1

2δ (δ <0), provided that

<

z0g⁰(z0) g(z₀)

6α.

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But the inequalities in (2.16) and (2.17) contradict the inequalities in (2.11). Hence|w(z)|<1, for allz ∈ U, and therefore (2.13) yields

(2.18)

z^−qf(z) g(z)

δ

−1

=|w(z)|<1,

which evidently implies (2.12), and this completes the proof of Theorem 2.3.

Theorem 2.4. Let δ ∈ R\ {0}; 0 6 α < p;n, m, p ∈ N; q = n −m;f(z) ∈ T(n), and g(z)∈ T(m).Iff(z)satisfies the inequality:

(2.19) <

1 + zf⁰⁰(z) f⁰(z)







< q+α+ _2δ¹ when δ >0 and g(z)∈ Km(α)

> q+α+ _2δ¹ when δ <0 and g(z)∈ K/ m(α), then

(2.20) <

(

z^−qmf⁰(z) ng⁰(z)

δ)

>0,

where the value of

z^{−q mf}_ng0⁰(z)^(z)

δ

is taken its principle value.

Proof. Letf(z)∈ T(n)andg(z)∈ T(m)withn−m∈ N.Since m f⁰(z)

n g⁰(z) =z^q+k₁z^q+1+k₂z^q+2+· · · ∈ T(q) (q=n−m∈ N), and if we definew(z)by

(2.21)

z^−qm f⁰(z) n g⁰(z)

δ

= 1 +w(z) (z ∈ U),

then by appealing to the same technique as in the proof of Theorem 2.3, we arrive at the assertion (2.20) of Theorem 2.4 under the conditions stated with (2.19).

3. SOME CONSEQUENCES OF MAINRESULTS

Among the various interesting and important consequences of Theorems 2.2 – 2.4, we mention now some of the corollaries relating to the classesT_λ(p;α), T_λ(α),S_p(α),K_p(α),S_p,K_p, S(α),K(α),which are easily deducible form the main results. Inequalities concerning analytic and multivalent functions were also studied in [2] – [6], and in [8] – [10].

Firstly, if we takeδ= 1, then Theorem 2.2 by virtue of (1.7) gives the following:

Corollary 3.1. Let a functionF(z)defined by (2.2) satisfy the condition:

<







1 +z_F00(z)

F⁰(z) − ^F_F⁰_(z)^(z) 1−p

F(z) zF⁰(z)







<1, (3.1)

(z ∈ U; 06α < p;p∈ N;f(z)∈ T(p)) thenf(z)∈ T_λ(p;α).

Next, if we takeδ−1 =λ= 0in Theorem 2.2, so thatF(z) =f(z),then we get

Corollary 3.2. IfF(z) =f(z)satisfies the condition in (3.1), thenf(z) ∈ S_p(α),i.e. f(z)is p−valent starlike of orderα(06α < p;p∈ N)inU.

If we takeδ=λ= 1in Theorem 2.2, so thatF(z) =zf⁰(z), then we obtain the following:

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Corollary 3.3. Iff(z)satisfies the condition

(3.2) <







1 +z

(zf⁰(z))⁰⁰

(zf⁰(z))⁰ −^(zf_zf⁰0^(z))(z)⁰

1−p

zf⁰(z) (zf⁰(z))⁰







<1 (z ∈ U; 06α < p;p∈ N),

thenf(z)∈ K_p(α),that isf(z)isp−valent convex of the orderα(06α < p;p∈ N)inU. Forp= 1in Corollaries 3.1 – 3.3 give the following:

Corollary 3.4. Let a functionF(z)defined by (2.2) satisfy the condition

<







1 +z_F00(z)

F⁰(z) − ^F_F⁰_(z)^(z) 1− _zF^F^(z)0(z)







<1, (3.3)

(z ∈ U; 06α <1; f(z)∈ T) thenf(z)∈ T_λ(α).

Corollary 3.5. If F(z) = f(z) satisfies the condition (3.3), then f(z) ∈ S(α), i.e. f(z) is starlike of orderα(06α <1)inU.

Corollary 3.6. Iff(z)satisfies the condition

(3.4) <







1 +z

(zf⁰(z))⁰⁰

(zf⁰(z))⁰ − ^(zf_zf⁰0^(z))(z)⁰

1− _(zf^zf₀⁰_(z))^(z)0







<1 (z ∈ U; 06α <1),

thenf(z)∈ K(α),i.e.,f(z)is convex of orderα(06α <1)inU. Let us takeδ= 1in Theorems 2.3 and 2.4, then we get the following:

Corollary 3.7. Let z ∈ U; 0 6 α < p; n, m, p ∈ N; f(z) ∈ T(n) and a function g(z) ∈ T(m)belong to the classS_m(α)withq=n−m ∈ N. Iff(z)satisfies the inequality:

(3.5) <

zf⁰(z) f(z)

< q+α+1 2, then

(3.6) <

z^−qf(z) g(z)

>0.

Corollary 3.8. Letz ∈ U; 06α < p; n, m, p∈ N; f(z)∈ T(n)and a functiong(z)inT(m) belong to the classK_m(α)withq=n−m ∈ N. Iff(z)satisfies the inequality:

(3.7) <

1 + zf⁰⁰(z) f⁰(z)

< q+α+ 1 2, then

(3.8) <

z^−qm f⁰(z) n g⁰(z)

>0.

Lastly, settingδ=−1in Theorems 2.3 and 2.4, we obtain the following:

Corollary 3.9. Let z ∈ U; 0 6 α < p; n, m, p ∈ N; f(z) ∈ T(n)and suppose a function g(z) ∈ T(m)does not belong to the classS_m(α)with q = n−m ∈ N. If f(z)satisfies the inequality:

(3.9) <

zf⁰(z) f(z)

> q +α− 1 2,

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then

(3.10) <

z^qg(z)

f(z)

>0.

Corollary 3.10. Let z ∈ U; 0 6 α < p; n, m, p ∈ N; f(z) ∈ T(n)and suppose a function g(z)inT(m)does not belong to the classKm(α)withq = n−m ∈ N. If f(z)satisfies the inequality:

(3.11) <

1 + zf⁰⁰(z) f⁰(z)

> q+α− 1 2, then

(3.12) <

z^q n g⁰(z) m f⁰(z)

>0.

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