Research Article Sufficient Conditions for Janowski Starlikeness

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1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 62925, 7 pages doi: /2007/62925 Research Article Sufficient Conditions for Janowski Starlikeness Rosihan M. Ali, V. Ravichandran, and N. Seenivasagan Received 10 March 2007; Revised 15 June 2007; Accepted 4 July 2007 Recommended by Teodor Bulboaca Let A,B,D,E [ 1,1] and let p(z) be an analytic function defined on the open unit disk, p(0) = 1. Conditions on A, B, D, ande aredeterminedsothat1+βzp (z) being subordinated to (1 + Dz)/(1 + Ez) implies that p(z)issubordinatedto(1+az)/(1 + Bz). Similar results are obtained by considering the expressions 1 + β(zp (z)/p(z)) and 1 + β(zp (z)/p 2 (z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike. Copyright 2007 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. Introduction Let be the class of all analytic functions defined in the open unit disk U :={z C : z < 1} andnormalizedbytheconditions f (0) = 0 = f (0) 1. Let S [A,B] denote the class of functions f satisfying the subordination zf (z) 1+Az, ( 1 B<A 1). (1.1) 1+Bz Functions in S [A,B] arecalledthejanowski starlike functions ([1, 2]). Certain wellknown subclasses of starlike functions are special cases of the class S [A,B] forsuitable choices of the parameters A and B. Forexample,when0 α<1, S [1 2α, 1] =: S α is the familiar class of starlike functions of order α and S [1 α,0] ={f : zf (z)/ 1 < 1 α (z U)} =: S (α). For 0 <α 1, let S [α, α] ={f : zf (z)/ 1 <α zf (z)/f(z)+1 (z U)}=: S [α]. Silverman [3], Obradowi c and Tuneski [4], and many others (see [5 9]) have studied properties of functions defined in terms of the quotient (1 + zf (z)/f (z))/(zf (z)/f(z)). In fact, Silverman [3] has obtained the order of starlikeness for the functions in the class

2 2 International Journal of Mathematics and Mathematical Sciences G b defined by { G b := f : 1+zf (z) / f } (z) zf (z) / 1 <b,0<b 1, z U (1.2) and Obradowi c and Tuneski [4] improved the result of Silverman [3] by showing G b S [0, b] S (2/( b)). Later, Tuneski [10] obtained conditions for the inclusion G b S [A,B]tohold.Ifweletzf (z)/f(z) =: p(z), then G b S [A,B]becomes 1+ zp (z) p(z) 2 1+Az 1+bz = p(z) 1+Bz. (1.3) Let f and 0 α<1. Frasin and Darus [11] have shown that ( zf(z) ) f (z) 2zf (z) (1 α)z 2 α = z2 f (z) f 2 (z) 1 < 1 α. (1.4) Again by writing z 2 f (z)/( ) 2 as p(z), we see that the above implication is special case of 1+β zp (z) p(z) 1+Dz 1+Az = p(z) 1+Ez 1+Bz. (1.5) Another special case of the above implications was considered by Ponnusamy and Rajasekaran [12]. Nunokawa et al. [13] have shown that if p(z) isanalyticinu, p(0) = 1and1+ zp (z) 1+z, thenp(z) 1+z. Using this, they have obtained a criterion for a normalized analytic function to be univalent. In this paper, we extend the result by replacing the subordinate function 1 + z by a function of the form (1 + Dz)/(1 + Ez). In fact, we determine conditions on A,B,D,E [ 1,1] so that 1+βzp (z) 1+Dz 1+Az = p(z) 1+Ez 1+Bz. (1.6) Similar results are obtained by considering the expressions 1 + β(zp (z)/p 2 (z)), 1+β(zp (z)/p(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike. 2. Differential subordination Lemma 2.1. Let 1 B<A 1, 1 E<D 1,andβ 0. Assume that (A B) β (D E) ( 1+B 2) + 2B(D E) Eβ(A B). (2.1)

3 Rosihan M. Ali et al. 3 If p(z) is analytic in U with p(0) = 1 and then Proof. Define the function P(z)by and the function w(z)by or equivalently by 1+βzp (z) 1+Dz 1+Ez, (2.2) p(z) 1+Az 1+Bz. (2.3) P(z):= 1+βzp (z), (2.4) w(z):= p(z) 1 A Bp(z), (2.5) p(z) = 1+Aw(z) 1+Bw(z). (2.6) Then w(z) is meromorphic in U and w(0) = 0. We need to show that w(z) < 1inU.By a computation, we get P(z) = ( 1+Bw(z) ) 2 +(A B)βzw (z) ( 1+Bw(z) ) 2. (2.7) Therefore P(z) 1 D EP(z) = (A B)βzw (z) (D E) ( 1+Bw(z) ) 2 E(A B)βzw (z). (2.8) Assume that there exists a point z 0 U such that max w(z) = w(z 0 ) = 1. (2.9) z z 0 Then by [14, Lemma 1.3, page 28], there exists k 1suchthatz 0 w (z 0 ) = kw(z 0 ). Let w(z 0 ) = e iθ. For this z 0,wehave P( ) z 0 1 D EP ( ) (A B)k β z = [ 0 I2 +(H J) 2 +4HJt 2 +4I(H + J)t ] 1/2 (A B)k β max 1 t 1 { [I2 +(H J) 2 +4HJt 2 +4I(H + J)t ] 1/2 }, (2.10)

4 4 International Journal of Mathematics and Mathematical Sciences where I := 2B(D E) kβe(a B), J := (D E)B 2, H := (D E), and t := cosθ.acomputation shows that P( ) z 0 1 D EP ( ) z 0 (A B) β k H + I + J. (2.11) Yet another calculation shows that the function ψ(k):= (A B) β k/(h + I + J) isan increasing function of k.sincek 1, we have ψ(k) ψ(1) and therefore P( ) z 0 1 D EP ( ) (A B) β z 0 (D E)(1 + B 2 )+ 2B(D E) Eβ(A B), (2.12) which by (2.1) is greater than or equal to 1. This contradicts P(z) (1 + Dz)/(1 + Ez)and completes the proof. Remark 2.2. When β = 1, E = 0 = B, andd = 1 = A, Lemma 2.1 reduces to [13, Lemma 1, page 1035]. Further if p(z) = z 2 f (z)/f(z) 2, Lemma 2.1 reduces to [13, Theorem1, page 1036]. By taking p(z) = zf (z)/f(z)inlemma 2.1, we have the following result. Theorem 2.3. Let the conditions of Lemma 2.1 hold. If f satisfies ( ) 1+β zf (z) 1+ zf (z) f (z) zf (z) 1+Dz 1+Ez, (2.13) then f S [A,B]. By taking β = 1, A = α = B, andd = E = δ (0 <α,δ 1) in Theorem 2.3, wehave the following result. Corollary 2.4. Let 0 <α 1 and δ = α/(1 + α) 2.If f satisfies ( ) ( ) zf (z) 1+ zf (z) f (z) zf (z) <δ 2+ zf (z) 1+ zf (z) f (z) zf (z), (2.14) then [α]. By taking β = 1 A = 1 2α, B = 1, D = (1 α)/2, and E = 0(0 α<1) in Theorem 2.3, we have the following result. Corollary 2.5. If f satisfies ( ) zf (z) 1+ zf (z) f (z) zf (z) < 1 α 3 (0 α<1), (2.15) then α. By replacing p(z) by1/p(z), β = 1, A by B, andb by A in Lemma 2.1, wehave the following result.

5 Rosihan M. Ali et al. 5 Lemma 2.6. Let 1 B<A 1, 1 E<D 1. Assume that (A B) (D E) ( 1+A 2) + E(A B) 2A(D E). (2.16) If p(z) is analytic in U with p(0) = 1 and then 1+ zp (z) p 2 (z) 1+Dz 1+Ez, (2.17) p(z) 1+Az 1+Bz. (2.18) When p(z) = zf (z)/f(z), in Lemma 2.6, we have the following theorem. Theorem 2.7. Let 1 B<A 1, 1 E<D 1. Assumethat(2.16) holds. If f satisfies 1+zf (z)/f (z) zf (z)/f(z) 1+Dz 1+Ez, (2.19) then f S [A,B]. Example 2.8. If f G 1 α/(2 α) 2 (0 α<1), then f S (α). If f satisfies 1+ zf (z) f (z) zf (z) <β 1+ zf (z) f (z) + zf (z) ( ) α β = 1+3α + α 2,0<α 1, (2.20) then f S [α]. Similarly if (2.20) holds with β = (1 α)/[1 + (1 2α) 2 + 5α 3 ] (0 α<1), then f S α. Remark 2.9. When E = 0andD = b (0 <b 1), Corollary 2.5 reduces to [10, Corollary 2.6, page 203]. When A = 0 = E and D = B = b (0 <b 1), Corollary 2.5 reduces to [4, Theorem 1, page 61]. When A = 0 = E and D = B = 1, Corollary 2.5 reduces to [3, Corollary 1, page 76]. Lemma Let 1 B<A 1, 1 E<D 1, AB 0, and β 0. Assume that β (A B) (D E)(1 + AB)+ (D E)(A + B) Eβ(A B). (2.21) Let p(z) be analytic in U with p(0) = 1 and then 1+β zp (z) p(z) 1+Dz 1+Ez, (2.22) p(z) 1+Az 1+Bz. (2.23) Proof. The proof is similar to the proof of Lemma 2.1.

6 6 International Journal of Mathematics and Mathematical Sciences Remark When Eβ 0, AB 0, Lemma 2.10 is valid provided the following conditions hold: (1 Aβ) 2{ 2Eβ(A + B)(D E) (A B) [ (D E) 2 +(Eβ) 2]} 4β 2 (A B)AB (2.24) instead of (2.21). Remark When β = 1, A = λ = E,andD = B = 0( λ 1), Lemma 2.10 reduces to [12, Theorem 1(iii), page 195]. Example By taking β = 1, B = 0, D = A/(1 + A), and E = 0inLemma 2.10,wehave the following result. Let 0 <A 1. Let p(z)beanalyticinu with p(0) = 1. If zp (z)/p(z) <A/(1 + A), then p(z) 1+Az.Whenp(z) = zf (z)/f(z), A = 1 α,wehavethefollowing result. If satisfies 1+ zf (z) f (z) zf (z) < 1 α 2 α (0 α<1), (2.25) then (α). By taking p(z) = z 2 f (z)/f 2 (z)inlemma 2.10, we have the following result. Theorem Let the conditions of Lemma 2.10 hold. If f satisfies ( ( ) ) zf(z) 1+β 2zf (z) 1+Dz f (z) 1+Ez, (2.26) then z 2 f (z) f 2 (z) 1+Az 1+Bz. (2.27) Remark When β = 1, A = α, B = 0, E = 0, and D = (1 α)/(2 α) (0 α<1), Theorem2.14 reduces to [11, Theorem 2.4, page 307]. Acknowledgment The authors gratefully acknowledge the support from the research Grant IRPA EAR. References [1] W. Janowski, Some extremal problems for certain families of analytic functions. I, Annales Polonici Mathematici, vol. 28, pp , [2] Y. Polatoǧlu 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, [3] H. Silverman, Convex and starlike criteria, International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 1, pp , 1999.

7 Rosihan M. Ali et al. 7 [4] M. Obradowič and N. Tuneski, On the starlike criteria defined by Silverman, Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka, no. 24, pp , [5] M. Nunokawa, S. Owa, H. Saitoh, A. Ikeda, and N. Koike, Some results for strongly starlike functions, Journal of Mathematical Analysis and Applications, vol. 212, no. 1, pp , [6] M. Nunokawa, S. Owa, H. Saitoh, and N. Takahashi, On a strongly starlikeness criteria, Bulletin of the Institute of Mathematics. Academia Sinica, vol. 31, no. 3, pp , [7] M. Obradović and S. Owa, On some criterions for starlikeness of order α, Rendiconti di Matematica e delle sue Applicazioni. Serie VII, vol. 8, no. 2, pp , [8] V. Ravichandran and M. Darus, On a criteria for starlikeness, International Mathematical Journal, vol. 4, no. 2, pp , [9] V. Ravichandran, M. Darus, and N. Seenivasagan, On a criteria for strong starlikeness, The Australian Journal of Mathematical Analysis and Applications, vol. 2, no. 1, pp. 1 12, 2005, article 6. [10] N. Tuneski, On the quotient of the representations of convexity and starlikeness, Mathematische Nachrichten, vol , no. 1, pp , [11] B. A. Frasin and M. Darus, On certain analytic univalent functions, International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 5, pp , [12] S. Ponnusamy and S. Rajasekaran, New sufficient conditions for starlike and univalent functions, Soochow Journal of Mathematics, vol. 21, no. 2, pp , [13] M. Nunokawa, M. Obradović, and S. Owa, One criterion for univalency, Proceedings of the American Mathematical Society, vol. 106, no. 4, pp , [14] S. Ruscheweyh, Convolutions in GeometricFunction Theory, vol. 83 of Seminar on Higher Mathematics, Presses de l Université de Montréal, Montreal, Quebec, Canada, Rosihan M. Ali: School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, Malaysia address: rosihan@cs.usm.my V. Ravichandran: School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, Malaysia address: vravi@cs.usm.my N. Seenivasagan: School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, Malaysia address: vasagan2000@yahoo.co.in

8 Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors Edson Denis Leonel, Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, Rio Claro, SP, Brazil ; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow , Russia; loskutov@chaos.phys.msu.ru Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Corporation