We apply the general theory of differential subordination to obtain certian interesting criteria for -valent starlikeness and strong starlikeness. Some applications of these results are also discussed

M. P. Jeyaraman

T. N. Shanmugam

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An Application of Differential Subordination

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Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2009, Article ID 685435, 13 pagesdoi:10.1155/2009/685435Research ArticleAn Application of Differential SubordinationT. N. Shanmugam1 and M. P. Jeyaraman21 Department of Mathematics, College of Engineering, Anna University, Chennai 600025, India2 Department of Mathematics, Easwari Engineering College, Chennai 600089, IndiaCorrespondence should be addressed to M. P. Jeyaraman, jeyaraman mp@yahoo.co.inReceived 27 July 2009; Accepted 8 October 2009Recommended by Narendra Kumar GovilWe apply the general theory of diﬀerential subordination to obtain certian interesting criteria forp-valent starlikeness and strong starlikeness. Some applications of these results are also discussed.Copyright q 2009 T. N. Shanmugam and M. P. Jeyaraman. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.1. IntroductionLet Ap  p ∈ N   {1,2,3,...}  be the class of functions f z  of the formf z    zp  ∞ m 1ap mzp m  1.1 which are analytic in the open unit disk Δ :  {z : |z| < 1}.Let P be the class of functions p z  of the formp z    1  ∞ n 1pnzn  1.2 which are analytic in Δ.I fp z  ∈Psatisﬁes Rp z  > 0  z ∈ Δ , then we say that p z  is aCarath´ eodory function.With a view to recalling the principle of subordination between analytic functions, letthe functions f and g be analytic in Δ. Then we say that the function f is subordinate to g ifthere exists a Schwarz function w z ,a n a l y t i ci nΔ withw 0    0, |w z | < 1  z ∈ Δ ,  1.3 2 International Journal of Mathematics and Mathematical Sciencessuch thatf z    g w z    z ∈ Δ .  1.4 We denote this subordination byf ≺ g, f z  ≺ g z   z ∈ Δ .  1.5 In particular, if the function g is univalent in Δ, the above subordination is equivalent tof 0    g 0  or f Δ  ⊂ g Δ .  1.6 For −1 ≤ b<a≤ 1a n d0<γ≤ 1, a function f ∈A p is said to be in the class S∗p γ,a,b if it satisﬁeszf  z f z ≺ p1   az1   bzγ.  1.7 Also, we write S∗p γ,1,−1  SS∗p γ , the class of strongly starlike p-valent functions of order γin Δ. S∗p 1,a,b  S∗p a,b , the class of Janowski starlike p-valent function, S∗p 1,−1  S∗p,t h eclass of p-valent starlike function, and S∗p 1 − 2γ,1  S∗p γ  0 ≤ γ<1 , the class of p-valentstarlike function of order γ.For Carath´ eodory functions, Miller  1  obtained certain suﬃcient conditions applyingthe diﬀerential inequalities. Recently, Nunokawa et al.  2  have given some improvementof result by Miller  1 . Recently Ravichandran and Jayamala  3  studied some subordinationresults for Carath´ eodory functions. In this paper by extending the result of Ravichandran andJayamala  3 ,w eﬁ n ds u ﬃcient conditions for the subordination p z  ≺ q z  to hold for givenq z  and criteria for p-valent starlikeness. Our results include results obtained by Nunokawaet al.  2 . We also give some criteria for p-valently starlikeness and strong starlikeness.To prove our result we need the following lemma due to Miller and Mocanu  4 .Lemma 1.1  see  4, Theorem3.4h, page 132  . Let q z  be analytic and univalent in the unit diskΔ and θ ω  and let φ ω  be analytic in a domain D containing q Δ  with φ w /  0 when w ∈ q Δ .SetQ z    zq  z φq z ,h  z    θq z   Q z .  1.8 Suppose that i  Q z  is starlike univalent in Δ, ii  R{zh  z /Q z }   R{θ  q z  /φ q z     zQ  z /Q z } > 0 for z ∈ Δ.If p z  is analytic in Δ with, p 0  q 0 , p Δ  ⊆ D, andθp z   zp  z φp z ≺ θq z   zq  z φq z ,  1.9 then p z  ≺ q z  and q z  is the best dominant.International Journal of Mathematics and Mathematical Sciences 32. Application of Differential SubordinationBy making use of Lemma 1.1, we ﬁrst prove the following theorem.Theorem 2.1. Let 0/  α ∈ C and λ be a positive real number. Let q z  be convex univalent in Δ andR  1 − α  \ α   m q z  m−1  > 0,m∈ N \{ 1}.I fp ∈Psatisﬁes 1 − α p z    αp z m   αλzp  z  ≺ h z ,  2.1 whereh z     1 − α q z    αq z m   αλzq  z ,  2.2 thenp z  ≺ q z ,  2.3 and q z  is the best dominant of  2.1 .Proof. Letθ w     1 − α w   αwm,φ  w    αλ.  2.4 Then clearly θ w  and φ w  are analytic in C and φ w /  0. Also letQ z    zq  z φq z   αλzq  z ,h z    θq z   Q z    1 − α q z    αq z m   αλzq  z . 2.5 Sinceq z isconvex univalent, zq  z isstarlike univalent. Therefore Q z isstarlikeunivalentin Δ,a n dRzh  z Q z  1λR1 − αα  mq z m−1   λ1  zq   z q  z > 0  2.6 for z ∈ Δ.From  2.1 – 2.6  we see thatθp z   zp  z φp z ≺ θq z   zq  z φq z   h z .  2.7 Therefore, by applying Lemma 1.1, we conclude that p z  ≺ q z  and q z  is the bestdominant of  2.1 . The proof of the theorem is complete.Bytakingαasrealandq z      1   az / 1   bz  γ inTheorem 2.1,wegetthefollowingcorollary.4 International Journal of Mathematics and Mathematical SciencesCorollary 2.2. Let −1 ≤ b<a≤ 1, m ∈ N \{ 1},0 <γ≤ 1/ m − 1 , λ be real number such thatλ>0 and 0 <α≤ 1.I fp ∈Psatisﬁes 1 − α p z    αp z m   αλzp  z  ≺ h z ,  2.8 whereh z     1 − α 1   az1   bzγ  α1   az1   bzmγ αλγ a − b z 1   az 1−γ 1   bz 1 γ ,  2.9 thenp z  ≺1   az1   bzγ,  2.10 and   1   az / 1   bz  γ is the best dominant of  2.8 .Corollary 2.3. Let −1 ≤ b<a≤ 1, λ>0.I ff ∈A p satisﬁes f z /  0 in 0 < |z| < 1 andzf  z pf z 	1 − α  αp1 − λpzf  z f z   αλ1  zf   z f  z ≺ h z ,  2.11 whereh z     1 − α 1   az1   bzγ  α1   az1   bz2γ αλγ a − b z 1   az 1−γ 1   bz 1 γ ,  2.12 thenzf  z pf z ≺1   az1   bzγ.  2.13 Proof. Let p z  zf  z /pf z , then p ∈Pand  2.11  can be written as 1 − α p z    αp2 z    αλzp  z ≺  1 − α 1   az1   bzγ  α1   az1   bz2γ αλγ a − b z 1   az 1−γ 1   bz 1 γ . 2.14 Taking m   2i nCorollary 2.2 and using  2.14 , we havezf  z pf z ≺1   az1   bzγ.  2.15 International Journal of Mathematics and Mathematical Sciences 5By taking p   λ   γ   a   1a n db   −1i nCorollary 2.3, we get the following result ofPadmanabhan  5 .Corollary 2.4. Let f ∈Aandzf  z f z   αz2f   z f  z ≺2αz2   2z  1 − z2 1 − z 2  0 <α≤ 1 ,  2.16 thenRzf  z f z > 0.  2.17 Theorem 2.5. Let α,β,ξ,η ∈ C and η/  0.L e tq z  be convex univalent in Δ and satisfyR	1ηβ   2ξq z > 0.  2.18 If p ∈Psatisﬁesα   βp z    ξp2 z    ηzp  z  ≺ α   βq z    ξq2 z    ηzq  z    h z ,  2.19 thenp z  ≺ q z ,  2.20 and q z  is the best dominant of  2.19 Proof. By setting θ w  :  α   βw   ξw2 and φ w  :  η it can be easily observed that θ w  andφ w  are analytic in C and that φ w /  0  w ∈ C \{ 0} .Also, by lettingQ z    zq  z φq z   ηzq  z ,h z    θq z   Q z   α   βq z    ξq2 z    ηzq  z , 2.21 we ﬁnd that Q z  is starlike univalent in Δ and thatRzh  z Q z   R	1ηβ   2ξq z  1  zq   z q  z > 0.  2.22 The diﬀerential subordinationα   βp z    ξp2 z    ηzp  z  ≺ α   βq z    ξq2 z    ηzq  z   2.23 6 International Journal of Mathematics and Mathematical Sciencesbecomesθp z   zp  z φp z ≺ θq z   zq  z φq z .  2.24 Now, the result follows as an application of Lemma 1.1.Theorem 2.6. Let α,β,ξ,η, and δ be complex numbers, δ/  0.L e t0/  q z  be univalent in Δ andsatisfy the following conditions for z ∈ Δ: 1  let Q z  δzq  z /q z  be starlike, 2  R{ β/δ q z    2ξ/δ q2 z  −  η/δq z     zQ  z /Q z } > 0.If p ∈Psatisﬁesα   βp z    ξ p z  2  ηp z   δzp  z p z ≺ α   βq z    ξ q z  2  ηq z   δzq  z q z ,  2.25 thenp z  ≺ q z ,  2.26 and q z  is the best dominant.Proof. The proof of this theorem is much akin to the proof of Theorem 2.5 and hence can beomitted.Remark 2.7. By taking α   β   0,ξ    λ/μ μ>0,λ>−μ/2,η   1, and q z    1   z / 1 − z in Theorem 2.5 we get the result of Nunokawa et al.  2  which was proved by a diﬀerentmethod.Remark 2.8. For the choices of α   β   0i nTheorem 2.5,w eg e tt h er e s u l to f 3, Theorem 1,page 192  and for α   ξ   η   0i nTheorem 2.6 we get the result of  3, Theorem 2, page 194 .Corollary 2.9. Let −1 ≤ b<a≤ 1,0 <γ≤ 1 and λ>0.I ff ∈A p satisﬁes f z f  z /  0 in0 < |z| < 1,t h e n 1 − λ zf  z f z   λ1  zf   z f  z ≺ p1   az1   bzγ λγ a − b z 1   az  1   bz  2.27 impliesf ∈ S∗pγ,a,b.  2.28 Also,1  zf   z f  z −zf  z f z ≺γ a − b z 1   az  1   bz  2.29 International Journal of Mathematics and Mathematical Sciences 7impliesf ∈ S∗pγ,a,b.  2.30 Proof. By taking α   ξ   η   0,β   p/λ, δ   1,p z  zf  z /pf z ,a n dq z    1   az / 1   bz  γ in Theorem 2.6,w eg e tt h eﬁ r s tp a r t .Proof of the second part follows, by setting α   β   ξ   η   0,δ   1,p z  zf  z /pf z ,a n dq z      1   az / 1   bz  γ.For α   ξ   0,β   1, p z  zf  z /f z ,a n dq z    1   az / 1 − z ,−1 <a≤ 1i nTheorem 2.5, we have the following result.Corollary 2.10. If f ∈Asatisﬁes f z /  0, z ∈ Δ andzf  z f z 	1 − ηzf  z f z   η1  zf   z f  z ≺ h z ,  2.31 whereh z   1   az1 − z  η 1   a z 1 − z 2 ,  2.32 thenzf  z f z ≺1   az1 − z.  2.33 One notes that if h z  u   iv,t h e nh Δ  is the exterior of the parabola given byv2   − 1   a η	u −2 − 2a − η 1   a 4 2.34 with its vertex as   2 − 2a − η 1   a /4 ,0  (see [5, 6]).By taking η   a   1i nCorollary 2.10, we obtain the following.Corollary 2.11. If f ∈Asatisﬁes f z /  0, z ∈ Δ, andzf  z f z 	2 −zf  z f z  zf   z f  z ≺1   2z − z2 1 − z 2 ,  2.35 thenzf  z f z ≺1   z1 − z.  2.36 Region h Δ  has been shown shaded in Figure 1.8 International Journal of Mathematics and Mathematical Sciencesv−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5u−3−2.5−2−1.5−1−0.50.511.522.53 η   A   1 Figure 1: η   a   1.Letting α   β   0,ξ   η   1,p z  zf  z /f z ,a n dq z    1    1 − 2γ z / 1 − z  inTheorem 2.5, we get the following.Corollary 2.12. If f ∈Asatisﬁes f z /  0,0 < |z| < 1, andzf  z f z 1  zf   z f  z ≺ h z ,  2.37 whereh z   1 − 2γ2z2   22 − 3γz   1 1 − z 2  2.38 for some γ  0 ≤ γ<1 ,t h e nRzf  z f z >γ .  2.39 For the univalent function h z  given by  2.38 , One now ﬁnds the image h Δ  of theunit disk Δ.International Journal of Mathematics and Mathematical Sciences 9Let h   u   iv, where u and v are real. One hasu   −2 − 3γ 1   2γ2 − 2γcosθ 1 − cosθ ,v  2γ1 − γsinθ1 − cosθ. 2.40 Elimination of θ yieldsv2   −8γ21 − γ3 − 2γu −2γ2   γ − 12.  2.41 Therefore, one concludes thath Δ   w   u   iv; v2 > −8γ21 − γ3 − 2γu −2γ2   γ − 12,  2.42 which properly contains the half plane Rw> 2γ2   γ − 1 /2.Corollary 2.13. Let −1 ≤ b<a≤ 1 and R β ≥ 0.I ff ∈A p satisﬁes f  z /  0 in 0 < |z| < 1 and1 − β f z zf  z  f z f   z f  z 2 ≺ h z ,  2.43 whereh z   bpb − βaz2  2p   1 − βb −1   βaz   p − βp 1   bz 2 ,  2.44 thenf ∈ S∗p b,a .  2.45 Proof. If we let p z  pf z /zf  z , then p ∈Pand  2.43  can be expressed asβp z    zp  z  ≺ β1   az1   bz  a − b z 1   bz 2.  2.46 Hence, by taking α   ξ   0,η   1,q z    1   az / 1   bz  and R β ≥ 0i nTheorem 2.5,w ehave p z  ≺  1   az / 1   bz .S o ,f z  ∈ S∗p b,a .Setting p   1a n db   −1i nCorollary 2.13, we get the following corollay.10 International Journal of Mathematics and Mathematical SciencesCorollary 2.14. Let −1 <a≤ 1 and R β ≥ 0.I ff ∈Asatisﬁes f  z /  0 in 0 < |z| < 1 and1 − β f z zf z  f z f   z f  z 2 ≺ h z ,  2.47 whereh z   1   βaz2  β − 3−1   βaz   1 − β 1 − z 2 ,  2.48 thenf z zf  z ≺1   az1 − z.  2.49 Remark 2.15. For the function h z  given by  2.48 , we haveh Δ   w   u   iv; v2 >a 0 u − b0 ,  2.50 which properly contains the half plane Rw>b 0, wherea0    1   a β2,b0  5   a   2β a − 1 4. 2.51 By putting p   a   β   1a n db   −1i nCorollary 2.13, we get the following result ofTuneski  7 .Corollary 2.16. If f z  ∈Aandf z f   z f  z 2 ≺2z z − 2  1 − z 2 ,  2.52 thenRf z zf  z > 0.  2.53 Remark 2.17. By putting 0   b<a≤ 1,p   1, and β   0i nCorollary 2.13, we get the resultobtained by Singh  8 , which reﬁnes the result of Silverman  9 .International Journal of Mathematics and Mathematical Sciences 11Corollary 2.18. Let 0/  η and q z  be convex univalent in Δ with q 0  1 and satisfy  2.18 .Let f ∈A p andψ z  :  α  βpf z zpμ ξp2f z zp2μ  ημf z zpμ	zf  z pf z − 1.  2.54 Ifψ z  ≺ α   βq z    ξq2 z    ηzq  z ,  2.55 then1pf z zpμ≺ q z ,  2.56 and q z  is the best dominant.Proof. By taking p z    1/p  f z /zp μ in Theorem 2.5, we have the above corollary.Corollary 2.19. Let 0/  λ ∈ C and q z  be convex univalent in Δ with q 0  1 and satisfyRμλ> 0.  2.57  i  If f ∈Asatisﬁes 1 − λ f z zμ  λf  z f z zμ−1≺ q z   λμzq  z ,  2.58 thenf z zμ≺ q z .  2.59  ii  If f ∈Asatisﬁesf  z f z zμ−1−f z zμ≺1μzq  z ,  2.60 thenf z zμ≺ q z ,  2.61 and q z  is the best dominant.12 International Journal of Mathematics and Mathematical SciencesProof. Proof of the ﬁrst part follows from Corollary 2.18, by taking β   p   1,α   ξ   0, andη   λ/μ.The proof of the second part follows from Corollary 2.18, by taking α   β   ξ   0,p   1,and η   1/μ.Bytakingλ   μ   nwherenisapositiveintegerandq z  A  1−A  −1− 2/z log 1−z   in the ﬁrst part of Corollary 2.19, we get the following result of Ponnusamy  10 .Corollary 2.20. Let f ∈A , then for a positive integer n, one has thatR 1 − n f z zn  nf  z f z zn−1>β  2.62 impliesf z zn≺ A    1 − A −1 −2zlog 1 − z ,  2.63 and A    1 − A  −1 −  2/z log 1 − z   is the best dominant.Remark 2.21. By taking μ   1a n dq z  1    A/ 1   δ  z in Corollary 2.19 and μ   λ   1a n dq z  A/B    1 − A/B  log 1   Bz /Bz  we get the result of Ponnusamy and Juneja  11 .By taking β   ξ   η   0,α   p   1, δ   1/μ,p z    1/p  f z /zp μ,a n dq z  eμAz inTheorem 2.5, we get the following result obtained by Owa and Obradovi´ c  12 .Corollary 2.22. Let f ∈Aandzf  z f z ≺ 1   Az,  2.64 thenf z zμ≺ eμAz,  2.65 and eμAz is the best dominant.We remark here that q z  eμAz is univalent if and only if |μA| <π.Remark 2.23. For a special case when p z    1/p  f z /zp μ,q z  1/ 1 − z 2b where b ∈C \{ 0} and β   ξ   η   0,α   μ   p   1, and δ   1/b in Theorem 2.6, we have the resultobtained by Srivastava and Lashin  13 .Corollary 2.24. If f ∈Asatisﬁes 1   λ zf z μ− λf  z zf z μ 1≺ q z   λμzq  z ,  2.66 International Journal of Mathematics and Mathematical Sciences 13thenzf z μ≺ q z ,  2.67 and q is the best dominant.Proof. By taking p z    1/p  zp/f z  μ and α   ξ   0,β   p   1a n dη   λ/μ in Theorem 2.5,we get the previous corollary.References 1  S. S. Miller, “Diﬀerential inequalities and Carath´ eodory functions,” Bulletin of the AmericanMathematical Society, vol. 81, pp. 79–81, 1975. 2  M. Nunokawa, S. Owa, N. Takahashi, and H. Saitoh, “Suﬃcient conditions for Carath´ eodoryfunctions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 9, pp. 1385–1390, 2002. 3  V. Ravichandran and M. Jayamala, “On suﬃcient conditions for Carath´ eodory functions,” Far EastJournal of Mathematical Sciences, vol. 12, no. 2, pp. 191–201, 2004. 4  S. S. Miller and P. T. Mocanu, Diﬀerential Subordinations: Theory and Applications, vol. 225 of Monographsand Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. 5  K. S. Padmanabhan, “On suﬃcient conditions for starlikeness,” Indian Journal of Pure and AppliedMathematics, vol. 32, no. 4, pp. 543–550, 2001. 6  S. Singh and S. 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Obradovi´ c, “An application of diﬀerential subordinations and some criteria forunivalency,” Bulletin of the Australian Mathematical Society, vol. 41, no. 3, pp. 487–494, 1990. 13  H. M. Srivastava and A. Y. Lashin, “Some applications of the Briot-Bouquet diﬀerential subordina-tion,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 2, pp. 1–7, 2005.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

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An Application of Differential Subordination

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