We consider the Fekete-Szegö problem with complex parameter  for the class () of analytic functions

Deepak Bansal

International Journal of Mathematics and Mathematical Sciences

Directory of Open Access Journals

Fekete-Szegö Problem for a New Class of Analytic Functions

We consider the Fekete-Szegö problem with complex parameter for the class () of analytic functions.</jats:p

Crossref

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2011, Article ID 143096, 5 pagesdoi:10.1155/2011/143096Research ArticleFekete-Szego¨ Problem for a New Class ofAnalytic FunctionsDeepak BansalDepartment of Mathematics, College of Engineering and Technology,Bikaner 334004, Rajasthan, IndiaCorrespondence should be addressed to Deepak Bansal, deepakbansal 79@yahoo.comReceived 2 December 2010; Revised 8 March 2011; Accepted 15 June 2011Academic Editor: Attila Gila´nyiCopyright q 2011 Deepak Bansal. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.We consider the Fekete-Szego¨ problem with complex parameter μ for the class Rτγ φ of analyticfunctions.1. Introduction and PreliminariesLetA denote the class of functions of the formfz  z ∞∑k2akzk, 1.1which are analytic in the open unit disk U  {z : z ∈ C and |z| < 1} and S denote the subclassofA that are univalent in U. A function fz inA is said to be in class S∗ of starlike functionsof order zero in U, if Rzf ′z/fz > 0 for z ∈ U. Let K denote the class of all functionsf ∈ A that are convex. Further, f is convex if and only if zf ′z is star-like. A function f ∈ Ais said to be close-to-convex with respect to a fixed star-like function g ∈ S∗ if and only ifRzf ′z/gz > 0 for z ∈ U. Let C denote of all such close-to-convex functions 1.Fekete and Szego¨ proved a noticeable result that the estimate∣∣∣a3 − λa22∣∣∣  1  2 exp( −2λ1 − λ)1.22 International Journal of Mathematics and Mathematical Sciencesholds for any normalized univalent function fz of the form 1.1 in the open unit disk Uand for 0  λ  1. This inequality is sharp for each λ see 2. The coeﬃcient functionalφλ(f) a3 − λa22 16(f ′′′0 − 3λ2[f ′′0]2), 1.3on normalized analytic functions f in the unit disk represents various geometric quantities,for example, when λ  1, φλf  a3 − a22, becomes Sf0/6, where Sf denote the Schwarzianderivative f ′′′/f ′′ −f ′′/f ′2/2 of locally univalent functions f in U. In literature, there existsa large number of results about inequalities for φλf corresponding to various subclasses ofS. The problem of maximising the absolute value of the functional φλf is called the Fekete-Szego¨ problem; see 2. In 3, Koepf solved the Fekete-Szego¨ problem for close-to-convexfunctions and the largest real number λ for which φλf is maximised by the Koebe functionz/1 − z2 is λ  1/3, and later in 4 see also 5, this result was generalized for functionsthat are close-to-convex of order β.Let φz be an analytic function with positive real part on U with φ0  1, φ′0 > 0which maps the unit disk U onto a star-like region with respect to 1 which is symmetric withrespect to the real axis. Let S∗φ be the class of functions in f ∈ S for whichzf ′zfz≺ φz z ∈ U, 1.4and Cφ be the class of functions in f ∈ S for which1 zf ′′zf ′z≺ φz z ∈ U, 1.5where ≺ denotes the subordination between analytic functions. These classes were introducedand studied by Ma and Minda 6. They have obtained the Fekete-Szego¨ inequality for thefunctions in the class Cφ.Motivated by the class Rτλβ in paper 7, we introduce the following class.Definition 1.1. Let 0  γ  1, τ ∈ C \ {0}. A function f ∈ A is in th class Rτγ φ, if1 1τ(f ′z  γzf ′′z − 1) ≺ φz z ∈ U, 1.6where φz is defined the same as above.If we setφz 1 Az1  Bz−1  B < A  1; z ∈ U, 1.7in 1.6, we getRτγ(1 Az1  Bz) Rτγ A,B {f ∈ A :∣∣∣∣∣f ′z  γzf ′′z − 1τA − B − B(f ′z  γzf ′′z − 1)∣∣∣∣∣ < 1}, 1.8International Journal of Mathematics and Mathematical Sciences 3which is again a new class. We list few particular cases of this class discussed in the literature1 Rτγ 1 − 2β,−1  Rτγ β for 0  β < 1, τ  C \ {0} was discussed recently bySwaminathan 7.2 The class Rτγ 1 − 2β,−1 for τ  eiη cos η, where −π/2 < η < π/2 is considered in8 see also 9.3 The class Rτ10,−1 with τ  eiη cos η was considered in 10 with reference to theunivalencey of partial sums.4 f ∈ Reiη cos ηγ 1 − 2β,−1 whenever zf ′z ∈ Pτγ β, the class considered in 11.For geometric aspects of these classes, see the corresponding references. The classRτγ A,B is new as the author Swaminathan 7 has introduced class Rτγ β which is subclassof the class Rτγ A,B, in his recent paper. To prove our main result, we need the followinglemma.Lemma 1.2 see 12, 13. If pz  1  c1z  c2z2  c3z3  · · · z ∈ U is a function with positivereal part, then for any complex number μ,∣∣∣c2 − μc21∣∣∣  2 max{1,∣∣2μ − 1∣∣}, 1.9and the result is sharp for the functions given bypz 1  z21 − z2 , pz 1  z1 − z z ∈ U. 1.102. Fekete-Szego¨ ProblemOur main result is the following theorem.Theorem 2.1. Let φz  1B1zB2z2 B3z3  · · · , where φz ∈ A with φ′0 > 0. If fz givenby 1.1 belongs to Rτγ φ0  γ  1, τ ∈ C \ {0}, z ∈ U, then for any complex number μ∣∣∣a3 − μa22∣∣∣  B1|τ |3(1  2γ) max{1,∣∣∣∣∣B2B1− 3τμB1(1  2γ)4(1  γ)2∣∣∣∣∣}. 2.1The result is sharp.Proof. If fz ∈ Rτγ φ, then there exists a Schwarz function wz analytic in U with w0  0and |wz| < 1 in U such that1 1τ(f ′z  γzf ′′z − 1)  φwz z ∈ U. 2.2Define the function p1z byp1z 1 wz1 −wz  1  c1z  c2z2  · · · . 2.34 International Journal of Mathematics and Mathematical SciencesSince wz is a Schwarz function, we see that Rp1z > 0 and p10  1. Define the functionpz bypz  1 1τ(f ′z  γzf ′′z − 1)  1  b1z  b2z2  · · · . 2.4In view of 2.2, 2.3, 2.4, we havepz  φ(p1z − 1p1z  1) φ(c1z  c2z2  · · ·2  c1z  c2z2  · · ·) φ(12c1z 12(c2 −c212)z2  · · ·) 1  B112c1z  B112(c2 −c212)z2  B214c21z2  · · · .2.5Thus,b1 12B1c1; b2 12B1(c2 −c212)14B2c21. 2.6From 2.4, we obtaina2 B1c1τ4(1  γ) ; a3 τ6(1  2γ)[B1(c2 −c212)12B2c21]. 2.7Therefore, we havea3 − μa22 B1τ6(1  2γ)(c2 − νc21), 2.8whereν 12(1 − B2B13τμB1(1  2γ)4(1  γ)2). 2.9Our result now is followed by an application of Lemma 1.2. Also, by the application ofLemma 1.2 equality in 2.1 is obtained whenp1z 1  z21 − z2 or p1z 1  z1 − z 2.10International Journal of Mathematics and Mathematical Sciences 5butpz  1 1τ(f ′z  γzf ′′z − 1)  φ(p1z−1p1z1). 2.11Putting value of p1z we get the desired results.For class Rτγ A,B,φz 1 Az1  Bz 1 Az1  Bz−1 z ∈ U 1  A − Bz −(AB − B2)z2  · · · .2.12Thus, putting B1  A − B and B2  −BA − B in Theorem 2.1, we get the following corollary.Corollary 2.2. If fz given by 1.1 belongs to Rτγ A,B, then∣∣∣a3 − μa22∣∣∣  A − B|τ |3(1  2γ) max{1,∣∣∣∣∣B 3τμA − B(1  2γ)4(1  γ)2∣∣∣∣∣}. 2.13References1 P. L. Duren, Univalent Functions, vol. 259 of Fundamental Principles of Mathematical Sciences, Springer,Berlin, Germany, 1983.2 M. Fekete and G. Szego¨, “Eine bemerkung uber ungerade schlichten funktionene,” Journal of LondonMathematical Society, vol. 8, pp. 85–89, 1993.3 W. Koepf, “On the Fekete-Szego¨ problem for close-to-convex functions,” Proceedings of the AmericanMathematical Society, vol. 101, no. 1, pp. 89–95, 1987.4 W. Koepf, “On the Fekete-Szego¨ problem for close-to-convex functions. II,”Archiv der Mathematik, vol.49, no. 5, pp. 420–433, 1987.5 R. R. London, “Fekete-Szego¨ inequalities for close-to-convex functions,” Proceedings of the AmericanMathematical Society, vol. 117, no. 4, pp. 947–950, 1993.6 W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” inProceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Lang, and S. Zhang, Eds., pp. 157–169, International Press, Cambridge, Mass, USA.7 A. Swaminathan, “Suﬃcient conditions for hypergeometric functions to be in a certain class ofanalytic functions,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1578–1583, 2010.8 S. Ponnusamy and F. Ronning, “Integral transforms of a class of analytic functions,” Complex Variablesand Elliptic Equations, vol. 53, no. 5, pp. 423–434, 2008.9 S. Ponnusamy, “Neighborhoods and Carathe´odory functions,” Journal of Analysis, vol. 4, pp. 41–51,1996.10 J. L. Li, “On some classes of analytic functions,”Mathematica Japonica, vol. 40, no. 3, pp. 523–529, 1994.11 A. Swaminathan, “Certain suﬃciency conditions on Gaussian hypergeometric functions,” Journal ofInequalities in Pure and Applied Mathematics, vol. 5, no. 4, article 83, 6 pages, 2004.12 F. R. Keogh and E. P. Merkes, “A coeﬃcient inequality for certain classes of analytic functions,”Proceedings of the American Mathematical Society, vol. 20, pp. 8–12, 1969.13 R. J. Libera and E. J. Złotkiewicz, “Coeﬃcient bounds for the inverse of a function with derivative inρ,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 251–257, 1983.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

http://downloads.hindawi.com/journals/ijmms/2011/143096.pdf

Fekete-Szegö Problem for a New Class of Analytic Functions

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Crossref