Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function using Quasi-Subordination

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1 Mathematica Aeterna, Vol. 3, 203, no. 3, Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function using Quasi-Subordination B. Srutha Keerthi Department of Applied Mathematics Sri Venkateswara College of Engineering Sriperumbudur, Chennai , India sruthilaya06@yahoo.co.in P. Lokesh Department of Mathematics Adhi Parasakthi College of Engineering G.B. Nagar, Kalavai , India lokeshpandurangan@gmail.com Abstract An analytic function f is quasi-subordinate to an analytic function g, in the open unit disk if there exist analytic functions ϕ and w, with ϕz), w0) = 0 and wz) < such that fz) = ϕz)gwz)). Certain subclass of analytic univalent functions associated with quasisubordination are defined and the bounds for the Fekete-Szegö coefficient functional a 3 µa 2 2 for functions belonging to these subclass is derived. Mathematics Subject Classification: 30C45 Keywords: Analytic functions, Univalent functions, Convex functions, Quasi-Subordination, Fekete-Szegö problem. Introduction and Motivation Let Abetheclassof analyticfunctionf intheopenunitdisk D = {z : z < } normalized by f0) = 0 and f 0) = of the form fz) = z + a n z n. For n=2

2 94 B. Srutha Keerthi and P. Lokesh two analytic functions f and g, the function f is subordinate to g, written as follows: fz) gz), ) if there exists an analytic function w, with w0) = 0 and wz) < such that fz) = gwz)). In particular, if the function g is univalent in D, then fz) gz) is equivalent to f0) = g0) and fd) gd). For brief survey on the concept of subordination, see []. Ma and Minda [2] introduced the following class { } S φ) = f A : zf z) fz) φz), 2) where φ is an analytic function with positive real part in D, φd) is symmetric withrespecttotherealaxisandstarlikewithrespecttoφ0) = andφ 0) > 0. A function f S φ) is called Ma-Minda starlike with respect to φ). The class Cφ) is the class of functions f A for which +zf z)/f z) φz). The class S φ) and Cφ) include several well-known subclasses of starlike and convex functions as special case. In the year 970, Robertson[3] introduced the concept of quasi-subordination. For two analytic functions f and g, the function f is quasi-subordinate to g, written as follows: fz) q gz), 3) if there exists analytic functions ϕ and w, with ϕz), w0) = 0 and wz) < such that fz) = ϕz)gwz)). Observe that when ϕz) =, then fz) = gwz)), so that fz) gz) in D. Also notice that if wz) = z, then fz) = ϕz)gz) and it is said that f is majorized by g and written fz) gz) in D. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. See [4, 5, 6] for works related to quasi-subordination. Throughout this paper it is assumed that φ is analytic in D with φ0) =. Motivated by [2, 3], we define the following class. Definition.. Let the class L q λ,φ), 0 λ ), consists of functions f A satisfying the quasi-subordination λz 3 f ++2λ)z 2 f +zf λz 2 f +zf q φz). 4) Example.2. The function f : D C defined by the following: belongs to the class L q λ,φ). λz 3 f ++2λ)z 2 f +zf λz 2 f +zf = zφz) ) 5)

3 Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function 95 It is well known see [0]) that the n-th coefficient of a univalent function f A is bounded by n. The bounds for coefficient give information about various geometric properties of the function. Many authors have also investigated the bounds for the Fekete-Szegö coefficient for various classes [, 2, 3, 4, 5, 6, 7, 8, 9, 20, 2, 22, 23, 24, 25]. In this paper, we obtain coefficient estimates for the functions in the above defined class. Let Ω be the class of analytic functions w, normalized by w0) = 0, and satisfying the condition wz) <. We need the following lemma to prove our results. Lemma.3. see [26]). If w Ω, then for any complex number t w 2 tw 2 max{; t }. 6) The result is sharp for the functions wz) = z 2 or wz) = z. 2 Main Results Throughout, let fz) = z+a 2 z 2 +a 3 z , φz) = + z+b 2 z 2 +B 3 z 3 +, ϕz) = c 0 +c z +c 2 z 2 +c 3 z , R and > 0. Theorem 2.. If f A belongs to L q λ,φ), 0 λ ), then a 2 a 3 and, for any complex number µ, a 3 µa λ) 2+λ), 6+2λ) +max{,b 2 + B 2 }), 7) +max { }), 3+2λ) 2+λ) 2µ B2 + B 2. 8) Proof. If f L q λ,φ), 0 λ ), then there exist analytic functions ϕ and w, with ϕz), w0) = 0 and wz) < such that Since λz 3 f ++2λ)z 2 f +zf λz 3 f ++2λ)z 2 f +zf λz 2 f +zf = ϕz)φwz)) ). 9) λz 2 f +zf = 2+λ)a 2 z + 4+λ) 2 a λ)a 3)z 2 +, φwz)) = w z + w 2 +B 2 w 2 )z 2 +, 0)

4 96 B. Srutha Keerthi and P. Lokesh ϕz)φwz)) ) = c 0 w z + c w +c 0 w 2 +B 2 w 2 ))z2 +, ) it follows from 9) that a 2 = c 0 w 2+λ), a 3 = 6+2λ) c w + c 0 w 2 +c 0 B 2 +B 2 c 0 )w 2 ). 2) Since ϕz) is analytic and bounded in D, we have [27, page 72] c n c 0 2 n > 0). 3) By using this fact and the well-known inequality, w, we get Further, a 3 µa 2 2 = 6+2λ) Then a 3 µa λ) a 2 2+λ). 4) c w +c 0 w 2 + B 2 +B 2 c 0 3+2λ) ) )) 2+λ) 2µB2 c 0 w 2. c w + c 0 w 2 Again applying c n and w, we have a 3 µa λ) w 2 3+2λ) 2+λ) 2µ Applying Lemma.3 to w 2 yields a 3 µa λ) 5) 3+2λ) 2+λ) 2µc 0 c 0 B 2 ) 3+2λ) ) 2+λ) 2µ c 0 B ) 2 { +max c 0 B ) 2 w 2 w 2 6) ). 7) 8), 3+2λ) ) 2+λ) 2µ c 0 B }) 2. 9) ) ) ) w 2.

5 Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function 97 Observe that 3+2λ) ) 2+λ) 2µ c 0 B 2 c 0 3+2λ) 2+λ) 2µ + B 2, 20) and hence we can conclude that a 3 µa λ) +max For µ = 0, the above will reduce to the estimate of a 3. Theorem 2.2. If f A satisfies λz 3 f ++2λ)z 2 f +zf then the following inequalities hold: { }), 3+2λ) 2+λ) 2µ B2 + B 2. 2) λz 2 f +zf φz), 22) a 2 a 3 and, for any complex number µ, a 3 µa λ) 2+λ), 6+2λ) +B 2 + B 2 ), 23) + ) 3+2λ) 2+λ) 2µ B2 + B 2. 24) Proof. The result follows by taking wz) = z in the proof of Theorem 2.. ACKNOWLEDGEMENT. The first author thanks the support provided by Science and Engineering Research Board DST), New Delhi. Project No: SR/S4/MS:76/0 with titled Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function using Quasi-Subordination. References [] P. Duren, Subordination, in Complex Analysis, Lecture Notes in Mathematics, Springer, Berlin, Germany, ), [2] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 992), Conference Proceedings Lecture Notes Analysis, International Press, Cambridge, Mass, USA, 994),

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