JJ II

(1)

volume 6, issue 1, article 9, 2005.

Received 01 May, 2004;

accepted 27 October, 2004.

Communicated by:N.K. Govil

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

SOME INEQUALITIES EXHIBITING CERTAIN PROPERTIES OF SOME SUBCLASSES OF MULTIVALENTLY ANALYTIC

FUNCTIONS

HÜSEYIN IRMAK AND R.K. RAINA

Department Of Mathematics Faculty Of Education Baskent University Tr-06530, Baglica Campus Baglica, Etimesgut- Ankara, Turkey.

EMail:hisimya@baskent.edu.tr Department Of Mathematics

College Of Technology and Engineering M.P. University Of Agriculture And Technology Udaipur- 31300, India

EMail:rainark_7@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 088-04

(2)

Some Inequalities Exhibiting Certain Properties Of Some Subclasses Of Multivalently

Analytic Functions Hüseyin Irmak and R.K. Raina

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of16

J. Ineq. Pure and Appl. Math. 6(1) Art. 9, 2005

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.

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

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

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.

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, . . .}),

which are analytic and multivalent in the open diskU ={z : z ∈ C and |z|<

1}. A functionf(z)belonging toT(p)is said to be multivalently starlike order αinU 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; 0 6α < p; p∈ N).

For the aforementioned definitions, one may refer to [1] (see also [11]). Fur- ther, a function f(z) ∈ T(p) is said to be in the subclass T SK^δ_λ(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; 06λ 61; 06α < p; p∈ N).

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of16

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 SK₀^δ(p;α) (δ6= 0, 06α < p, p∈ N), (1.5)

T K^δ(p;α)≡ T SK₁^δ(p;α) (δ6= 0, 06α < p, p∈ N), (1.6)

T_λ(p;α)≡ T SK_λ¹(p;α) (0 6λ61, 06α < p, p∈ N)(see [5]).

(1.7)

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) inU, multivalently starlike functionsSpinU, multivalently convex functionsKpinU, starlike functions S(α)of order α (0 6 α < 1) inU, convex functions K(α)of order α (0 6 α < 1)inU, starlike functions S inU and convex functions KinU, are seen to be easily identifiable with the aforementioned classes ([1], [5] and [11]).

By introducing a subclassT SK_λ^δ(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.

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of16

2. Main Results

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

Lemma 2.1 ([7]). Let the functionw(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 pointz0,then

(2.1) z₀w⁰(z₀) =c w(z₀) (c>1).

We begin now to prove the following:

Theorem 2.2. Let δ ∈ R\ {0},0 6 α < p, p ∈ N and f(z) ∈ T(p). If a functionF(z)defined by

(2.2) F(z) = (1−λ)f(z) +λzf⁰(z) (06λ 61), 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

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of16

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)akz^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 function w(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).

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of16

Therefore, (2.7) yields

<





 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)

(2.8)

= 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,

and hencef(z)∈ T SK_λ^δ(p;α).

Theorem 2.3. Letδ ∈R\ {0}; 06α < 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),

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of16

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 that z₀w⁰(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)

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of16

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.

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. Letδ ∈ 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:

(2.19) <

1 + zf⁰⁰(z) f⁰(z)







< q+α+ _2δ¹ whenδ >0andg(z)∈ K_m(α)

> q+α+ _2δ¹ whenδ <0andg(z)∈ K/ _m(α),

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of16

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).

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of16

3. Some Consequences of Main Results

Among the various interesting and important consequences of Theorems 2.2– 2.4, we mention now some of the corollaries relating to the classes Tλ(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 take δ = 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 Theorem2.2, so that F(z) = f(z),then we get

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

If we takeδ =λ= 1in Theorem2.2, so thatF(z) =zf⁰(z), then we obtain the following:

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of16

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),

then f(z) ∈ K_p(α),that is f(z)isp−valent convex of the order α (0 6 α <

p;p∈ N)inU.

Forp= 1in Corollaries3.1–3.3give 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. IfF(z) = f(z)satisfies the condition (3.3), thenf(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.

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of16

Let us takeδ= 1in Theorems2.3and2.4, 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 functiong(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. Let z ∈ U; 0 6 α < 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 Theorems2.3and2.4, we obtain the following:

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of16

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 classSm(α)withq = n−m∈ N. Iff(z)satisfies the inequality:

(3.9) <

zf⁰(z) f(z)

> q+α− 1 2,

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 functiong(z)inT(m)does not belong to the classK_m(α)withq = n−m∈ N. Iff(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.

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of16

References

[1] P.L. DUREN, Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer- Verlag, New York Berlin, Heidelberg, and Tokyo, 1983.

[2] H. IRMAKANDS. OWA, Certain inequalities involving analytic and uni- valent functions, Far East J. Math. Sci., 10 (2003), 353–358.

[3] H. IRMAK AND S. OWA, Certain inequalities for multivalent and mero- morphically multivalent starlike functions, Bull. Inst. Math. Acad. Sinica, 31 (2003), 11–21.

[4] H. IRMAK , R.K. RAINAANDS. OWA, A certain for multivalent starlike and mermorphically multivalent starlike functions, Inter. J. Appl. Math., 12 (2003), 93–98.

[5] H. IRMAKANDR.K. RAINA, The starlikeness and convexity of multiva- lent function involving certain inequalities, Rev. Mat. Complut.,16 (2003), 391–398.

[6] H. IRMAK, R.K. RAINAANDS. OWA, Certain results involving inequal- ities on analytic and univalent functions, Far East J. Math. Sci., 10 (2003), 359–366.

[7] I.S. JACK, Functions starlike and convex of the orderα,J. London Math.

Soc., 3 (1971), 469–474.

[8] S. OWA, M. NUNOKAWAANDH. SAITOH, Some inequalities involving multivalent functions, Ann. Polon. Math., 60 (1994), 159–162.

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of16

[9] S. OWA, M. NUNOKAWAANDS. FUKUI, A criterion forp−valent star- like functions, Intern. J. Math. & Math. Sci., 17 (1994), 205–207.

[10] S. OWA, H.M. SRIVASTAVA, F.-Y. RENANDW.-Q. YANG, The starlike- ness of a certain class of integral operators, Complex Variables, 27 (1995), 185–191.

[11] H.M. SRIVASTAVA AND S. OWA, (EDITORS), Current Topics in Ana- lytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.