Subordination for Higher-Order Derivatives of Multivalent Functions

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Volume 2008, Article ID 830138,12pages doi:10.1155/2008/830138

Research Article

Subordination for Higher-Order Derivatives of Multivalent Functions

Rosihan M. Ali,¹Abeer O. Badghaish,^{1, 2} and V. Ravichandran³

1School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia

2Mathematics Department, King Abdul Aziz University, P.O. Box 581, Jeddah 21421, Saudi Arabia

3Department of Mathematics, University of Delhi, Delhi 110 007, India

Correspondence should be addressed to Rosihan M. Ali,[email protected] Received 18 July 2008; Accepted 24 November 2008

Recommended by Vijay Gupta

Diﬀerential subordination methods are used to obtain several interesting subordination results and best dominants for higher-order derivatives ofp-valent functions. These results are next applied to yield various known results as special cases.

Copyrightq2008 Rosihan M. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Motivation and preliminaries

For a fixedp∈N:{1,2, . . .}, letApdenote the class of all analytic functions of the form

fz z^p^∞

k1

a_kpz^kp, 1.1

which are p-valent in the open unit disc U {z ∈ C : |z| < 1} and let A : A1. Upon diﬀerentiating both sides of1.1q-times with respect toz, the following diﬀerential operator is obtained:

f^qz λp;qz^p−q^∞

k1

λkp;qa_kpz^kp−q, 1.2

where

λp;q: p!

p−q!

p≥q;p∈N;q∈N∪ {0}

. 1.3

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Several researchers have investigated higher-order derivatives of multivalent functions, see, for example,1–10. Recently, by the use of the well-known Jack’s lemma11,12, Irmak and Cho5obtained interesting results for certain classes of functions defined by higher-order derivatives.

Letfandgbe analytic inU. Thenfissubordinatetog, written asfz≺gz z∈U if there is an analytic functionwzwithw0 0 and|wz|<1,such thatfz gwz.

In particular, if g is univalent in U, then f subordinate to g is equivalent to f0 g0 and fU ⊆ gU. A p-valent function f ∈ Ap is starlike if it satisfies the condition 1/pRzfz/fz > 0z ∈ U. More generally, let φz be an analytic function with positive real part inU, φ0 1, φ0>0,andφzmaps the unit discUonto a region starlike with respect to 1 and symmetric with respect to the real axis. The classesS^∗_pφandC_pφ consist, respectively, ofp-valent functionsf starlikewith respect toφandp-valent functions fconvexwith respect toφinUgiven by

f∈S^∗_pφ⇐⇒ 1 p

zfz

fz ≺φz, f∈Cpφ⇐⇒ 1 p

1zfz fz

≺φz. 1.4

These classes were introduced and investigated in 13, and the functions hφ,p and kφ,p, defined, respectively, by

1 p

zh_φ,p

h_φ,p φz

z∈U, hφ,p∈ Ap , 1

p

1zk_φ,p k_φ,p

φz

z∈U, kφ,p∈ Ap ,

1.5

are important examples of functions inS^∗_pφandC_p^∗φ. Ma and Minda14have introduced and investigated the classesS^∗φ:S^∗₁φandCφ:C₁φ. For−1≤B < A≤1, the class S^∗A, B S^∗1Az/1Bzis the class of Janowski starlike functionscf.15,16.

In this paper, corresponding to an appropriate subordinate functionQzdefined on the unit disk U, suﬃcient conditions are obtained for a p-valent function f to satisfy the subordination

f^qz

λp;qz^p−q ≺Qz, zf^q1z

f^qz −pq1≺Qz. 1.6

In the particular case when q 1 and p 1, and Qz is a function with positive real part, the first subordination gives a suﬃcient condition for univalence of analytic functions, while the second subordination implication gives conditions for convexity of functions. If q 0 and p 1, the second subordination gives conditions for starlikeness of functions.

Thus results obtained in this paper give important information on the geometric prop- erties of functions satisfying diﬀerential subordination conditions involving higher-order derivatives.

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The following lemmas are needed to prove our main results.

Lemma 1.1see12, page 135, Corollary 3.4h.1. LetQbe univalent inU, andϕbe analytic in a domainDcontainingQU. IfzQz·ϕQzis starlike, andPis analytic inUwithP0 Q0 andPU⊂D, then

zPz·ϕ Pz

≺zQz·ϕ Qz

⇒P ≺Q, 1.7

andQis the best dominant.

Lemma 1.2see12, page 135, Corollary 3.4h.2. LetQbe convex univalent inU, and letθbe analytic in a domainDcontainingQU. Assume that

R θQz 1zQz Qz

>0. 1.8

IfPis analytic inUwithP0 Q0andPU⊂D, then zPz θ

Pz

≺zQz θ Qz

⇒P ≺Q, 1.9

2. Main results

The first four theorems below give suﬃcient conditions for a diﬀerential subordination of the form

f^qz

λp;qz^p−q ≺Qz 2.1

to hold.

Theorem 2.1. LetQzbe univalent and nonzero inU,Q0 1, and letzQz/Qzbe starlike inU. If a functionf ∈ Apsatisfies the subordination

zf^q1z

f^qz ≺ zQz

Qz p−q, 2.2

then

f^qz

λp;qz^p−q ≺Qz, 2.3

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Proof. Define the analytic functionPzby

Pz: f^qz

λp;qz^p−q. 2.4

Then a computation shows that

zf^q1z

f^qz zPz

Pz p−q. 2.5

The subordination2.2yields zPz

Pz p−q≺ zQz

Qz p−q, 2.6

or equivalently

zPz

Pz ≺ zQz

Qz . 2.7

Define the function ϕ by ϕw : 1/w. Then 2.7 can be written as zPz·ϕPz ≺ zQz·ϕQz. Since Qz/0, ϕw is analytic in a domain containing QU. Also zQz·ϕQz zQz/Qzis starlike. The result now follows fromLemma 1.1.

Remark 2.2. Forf ∈ Ap, Irmak and Cho5, page 2, Theorem 2.1showed that

Rzf^q1z

f^qz < p−q⇒f^qz< λp;q|z|^p−q−1. 2.8 However, it should be noted that the hypothesis of this implication cannot be satisfied by any function inApas the quantity

zf^q1z f^qz

z0

p−q. 2.9

Theorem 2.1is the correct formulation of their result in a more general setting.

Corollary 2.3. Let−1≤B < A≤1. Iff∈ Apsatisfies zf^q1z

f^qz ≺ zA−B

1Az1Bzp−q, 2.10

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then

f^qz

λp;qz^p−q ≺ 1Az

1Bz. 2.11

Proof. For−1≤B < A≤1, define the functionQby Qz 1Az

1Bz. 2.12

Then a computation shows that

Fz: zQz

Qz A−Bz

1Az1Bz, hz: zFz

Fz 1−ABz² 1Az1Bz.

2.13

Withzre^iθ, note that R

h re^iθ

R 1−ABr²e^2iθ 1Are^iθ1Bre^iθ

1−ABr²1ABr² ABrcosθ

|1Are^iθ1Bre^iθ|² .

2.14

Since 1ABr² ABrcosθ≥1−Ar1−Br>0 forAB≥0, and similarly, 1ABr² ABrcosθ ≥ 1Ar1Br> 0 forAB ≤0, it follows thatRhz > 0,and hence zQz/Qzis starlike. The desired result now follows fromTheorem 2.1.

Example 2.4. 1For 0< β <1, chooseAβandB0 inCorollary 2.3. Sincew≺βz/1βz is equivalent to|w| ≤β|1−w|,it follows that iff∈ Apsatisfies

zf^q1z

f^qz −pq β² 1−β²

< β

1−β², 2.15

then

f^qz λp;qz^p−q −1

< β. 2.16

2WithA1 andB0,it follows fromCorollary 2.3that wheneverf ∈ Apsatisfies

R

zf^q1z f^qz −pq

< 1

2, 2.17

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then

f^qz λp;qz^p−q −1

<1. 2.18

Takingq0 andQz hφ,p/z^p,Theorem 2.1yields the following corollary.

Corollary 2.5see13. Iff ∈S^∗_pφ,then

fz z^p ≺ hφ,p

z^p . 2.19

Similarly, choosing q 1 and Qz k_φ,p /pz^p−1,Theorem 2.1yields the following corollary.

Corollary 2.6see13. Iff ∈C_p^∗φ,then

fz z^p−1 ≺ k_φ,p

z^p−1. 2.20

Theorem 2.7. LetQzbe convex univalent inUandQ0 1. Iff∈ Apsatisfies f^qz

λp;qz^p−q·

zf^q1z f^qz −pq

≺zQz, 2.21

then

f^qz

λp;qz^p−q ≺Qz, 2.22

Proof. Define the analytic functionPzbyPz :f^qz/λp;qz^p−q.Then it follows from 2.5that

f^qz λp;qz^p−q·

zf^q1z f^qz −pq

zPz. 2.23

By assumption, it follows that

zPz·ϕ Pz

≺zQz·ϕ Qz

, 2.24

whereϕw 1. SinceQzis convex, andzQz·ϕQz zQzis starlike,Lemma 1.1 gives the desired result.

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Example 2.8. When

Qz:1 z

λp;q, 2.25

Theorem 2.7is reduced to the following result in5, page 4, Theorem 2.4. Forf∈ Ap, f^qz·

zf^q1z

f^qz −pq

≤ |z|^p−q⇒f^qz−λp;qz^p−q≤ |z|^p−q. 2.26 In the special caseq 1, this result gives a suﬃcient condition for the multivalent function fzto be close-to-convex.

Theorem 2.9. LetQzbe convex univalent inUandQ0 1. Iff∈ Apsatisfies zf^q1z

λp;qz^p−q ≺zQz p−qQz, 2.27

then

f^qz

λp;qz^p−q ≺Qz, 2.28

Proof. Define the functionPzbyPz f^qz/λp;qz^p−q.It follows from2.5that zPz p−qPz≺zQz p−qQz, 2.29

that is,

zPz θ Pz

≺zQz θ Qz

, 2.30

where θw p−qw. The conditions in Lemma 1.2are clearly satisfied. Thus f^qz/

λp;qz^p−q≺Qz,andQis the best dominant.

Takingq0,Theorem 2.9yields the following corollary.

Corollary 2.10see17, Corollary 2.11. LetQzbe convex univalent inU,andQ0 1. If f∈ Apsatisfies

fz

z^p−1 ≺zQz pQz, 2.31

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then

fz

z^p ≺Qz. 2.32

Withp1,Corollary 2.10yields the following corollary.

Corollary 2.11 see17, Corollary 2.9. LetQzbe convex univalent inU,and Q0 1. If f∈ Asatisfies

fz≺zQz Qz, 2.33

then

fz

z ≺Qz. 2.34

Theorem 2.12. LetQzbe univalent and nonzero inU,Q0 1, andzQz/Q²zbe starlike.

Iff∈ Apsatisfies

λp;qz^p−q f^qz ·

zf^q1z f^qz −pq

≺ zQz

Q²z, 2.35

then

f^qz

λp;qz^p−q ≺Qz, 2.36

Proof. Define the functionPzbyPz f^qz/λp;qz^p−q.It follows from2.5that λp;qz^p−q

f^qz ·

zf^q1z

f^qz −p−q

1

Pz·zPz

Pz zPz

P²z. 2.37

By assumption,

zPz

P²z ≺ zQz

Q²z. 2.38

Withϕw:1/w²,2.38can be written aszPz·ϕPz≺zQz·ϕQz.The function ϕwis analytic inC− {0}.SincezQzϕQzis starlike, it follows fromLemma 1.1that Pz≺Qz,andQzis the best dominant.

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The next four theorems give suﬃcient conditions for the following diﬀerential subordination

zf^q1z

f^qz −pq1≺Qz 2.39

to hold.

Theorem 2.13. Let Qz be univalent and nonzero in U, Q0 1, Qz/q− p 1, and zQz/QzQz p−q−1be starlike inU. Iff∈ Apsatisfies

1 zf^q2z/f^q1z−pq1

zf^q1z/f^qz−pq1 ≺1 zQz

QzQz p−q−1, 2.40

then

zf^q1z

f^qz −pq1≺Qz, 2.41

Proof. Let the functionPzbe defined by

Pz zf^q1z

f^qz −pq1. 2.42

Upon diﬀerentiating logarithmically both sides of2.42, it follows that zPz

Pz p−q−1 1zf^q2z

f^q1z −zf^q1z

f^qz . 2.43

Thus

1zf^q2z

f^q1z −pq1 zPz

Pz p−q−1 Pz. 2.44

The equations2.42and2.44yield

1 zf^q2z/f^q1z−pq1

zf^q1z/f^qz−pq−1 zPz

PzPz p−q−11. 2.45

Iff∈ Apsatisfies the subordination2.40,2.45gives zPz

PzPz p−q−1 ≺ zQz

QzQz p−q−1, 2.46

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that is,

zPz·ϕ Pz

≺zQz·ϕ Qz

2.47 withϕw:1/wwp−q−1.The desired result is now established by an application of Lemma 1.1.

Theorem 2.13 contains a result in 18, page 122, Corollary 4 as a special case. In particular, we note that Theorem 2.13with p 1, q 0, and Qz 1Az/1 Bz for−1≤B < A≤1 yields the following corollary.

Corollary 2.14see18, page 123, Corollary 6. Let−1≤B < A≤1. Iff∈ Asatisfies 1 zfz/fz

zfz/fz ≺1 A−Bz

1Az², 2.48

thenf ∈S^∗A, B.

ForA0, BbandA1, B−1,Corollary 2.14gives the results of Obradoviˇc and Tuneski19.

Theorem 2.15. Let Qz be univalent and nonzero in U, Q0 1, Qz/q−p1, and let zQz/Qz p−q−1be starlike inU. Iff ∈ Apsatisfies

1zf^q2z

f^q1z −zf^q1z

f^qz ≺ zQz

Qz p−q−1, 2.49

then

zf^q1z

f^qz −pq1≺Qz, 2.50

Proof. Let the functionPzbe defined by 2.42. It follows from2.43and the hypothesis that

zPz

Pz p−q−1 ≺ zQz

Qz p−q−1. 2.51

Define the functionϕbyϕw:1/wp−q−1.Then2.51can be written as zPz·ϕ

Pz

≺zQz·ϕ Qz

. 2.52

Sinceϕwis analytic in a domain containingQU, andzQz·ϕQzis starlike, the result follows fromLemma 1.1.

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Theorem 2.16. LetQzbe a convex function inU,andQ0 1. Iff ∈ Apsatisfies zf^q1z

f^qz 2zf^q2z

f^q1z −zf^q1z f^qz

≺zQz Qz p−q−1, 2.53

then

zf^q1z

f^qz −pq1≺Qz, 2.54

Proof. Let the functionPzbe defined by2.42. Using2.43, it follows that zf^q1z

f^qz

1zf^q2z

zPz, 2.55

and, therefore,

zf^q1z f^qz

2zf^q2z

f^q1z − zf^q1z f^qz

zPz Pz p−q−1. 2.56

By assumption,

zPz Pz p−q−1≺zQz Qz p−q−1, 2.57

or

zPz θ Pz

≺zQz θ Qz

, 2.58

where the functionθw wp−q1.The proof is completed by applyingLemma 1.2.

Theorem 2.17. LetQzbe a convex function inU, withQ0 1. Iff ∈ Apsatisfies zf^q1z

f^qz 1zf^q2z

≺zQz, 2.59

then

zf^q1z

f^qz −pq1≺Qz, 2.60

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Proof. Let the functionPzbe defined by2.42. It follows from2.43thatzPz·ϕPz≺ zQz·ϕQz,whereϕw 1.The result follows easily fromLemma 1.1.

Acknowledgment

This work was supported in part by the FRGS and Science Fund research grants, and was completed while the third author was visiting USM.

References

1 O. Altıntas¸, “Neighborhoods of certain p-valently analytic functions with negative coeﬃcients,”

Applied Mathematics and Computation, vol. 187, no. 1, pp. 47–53, 2007.

2 O. Altıntas¸, H. Irmak, and H. M. Srivastava, “Neighborhoods for certain subclasses of multivalently analytic functions defined by using a diﬀerential operator,”Computers & Mathematics with Applications, vol. 55, no. 3, pp. 331–338, 2008.

3 M. P. Chen, H. Irmak, and H. M. Srivastava, “Some multivalent functions with negative coeﬃcients defined by using a diﬀerential operator,”Panamerican Mathematical Journal, vol. 6, no. 2, pp. 55–64, 1996.

4 H. Irmak, “A class ofp-valently analytic functions with positive coeﬃcients,”Tamkang Journal of Mathematics, vol. 27, no. 4, pp. 315–322, 1996.

5 H. Irmak and N. E. Cho, “A diﬀerential operator and its applications to certain multivalently analytic functions,”Hacettepe Journal of Mathematics and Statistics, vol. 36, no. 1, pp. 1–6, 2007.

6 H. Irmak, S. H. Lee, and N. E. Cho, “Some multivalently starlike functions with negative coeﬃcients and their subclasses defined by using a diﬀerential operator,”Kyungpook Mathematical Journal, vol. 37, no. 1, pp. 43–51, 1997.

7 M. Nunokawa, “On the multivalent functions,”Indian Journal of Pure and Applied Mathematics, vol. 20, no. 6, pp. 577–582, 1989.

8 Y. Polato ˘glu, “Some results of analytic functions in the unit disc,” Publications de l’Institut Mathematique, vol. 78, no. 92, pp. 79–85, 2005.

9 H. Silverman, “Higher order derivatives,”Chinese Journal of Mathematics, vol. 23, no. 2, pp. 189–191, 1995.

10 T. Yaguchi, “The radii of starlikeness and convexity for certain multivalent functions,” in Current Topics in Analytic Function Theory, pp. 375–386, World Scientific, River Edge, NJ, USA, 1992.

11 I. S. Jack, “Functions starlike and convex of orderα,”Journal of the London Mathematical Society. Second Series, vol. 3, pp. 469–474, 1971.

12 S. S. Miller and P. T. Mocanu,Diﬀerential Subordinations: Theory and Application, vol. 225 ofMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

13 R. M. Ali, V. Ravichandran, and S. K. Lee, “Subclasses of multivalent starlike and convex functions,”

to appear inBulletin of the Belgian Mathematical Society - Simon Stevin.

14 W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,”

inProceedings of the Conference on Complex Analysis, Conference Proceedings and Lecture Notes in Analysis, I, pp. 157–169, International Press, Tianjin, China, 1994.

15 W. Janowski, “Some extremal problems for certain families of analytic functions. I,”Annales Polonici Mathematici, vol. 28, pp. 297–326, 1973.

16 Y. Polato ˘glu and M. Bolcal, “The radius of convexity for the class of Janowski convex functions of complex order,”Matematichki Vesnik, vol. 54, no. 1-2, pp. 9–12, 2002.

17 O. ¨¨ O. Kılıc¸, “Suﬃcient conditions for subordination of multivalent functions,”Journal of Inequalities and Applications, vol. 2008, Article ID 374756, 8 pages, 2008.

18 V. Ravichandran and M. Darus, “On a criteria for starlikeness,”International Mathematical Journal, vol.

4, no. 2, pp. 119–125, 2003.

19 M. Obradowiˇc and N. Tuneski, “On the starlike criteria defined by Silverman,”Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka, vol. 181, no. 24, pp. 59–64, 2000.