On a subclass of analytic functions involving harmonic means

DOI: 10.1515/auom-2015-0018 An. Şt. Univ. Ovidius Constanţa Vol. 231),2015, 267 275 On a subclass of analytic functions involving harmonic means Andreea-Elena Tudor Dorina Răducanu Abstract In the present paper, we consider a generalised subclass of analytic functions involving arithmetic, geometric harmonic means. For this function class we obtain an inclusion result, Fekete-Szegö inequality coefficient bounds for bi-univalent functions. 1 Introduction Let U r = {z C : z < r} r > 0) let U = U 1 denote the unit disk. Let A be the class of all analytic functions f in U of the form: f = z + a n z n, z U. 1) n=2 Further, by S we shall denote the class of all functions in A which are { univalent} in U. It is known see 6]) that if f S, then fu) contains the disk w < 1 4. Here 1 4 is the best possible constant known as the Koebe constant for S. Thus every univalent function f has an inverse f 1 defined on some disk containing the disk { w < 4} 1 satisfying: f 1 f) = z, z U ff 1 w)) = w, w < r 0 f), r 0 f) 1 4, Key Words: Analytic functions, Fekete-Szegö inequality, bi-univalent functions. 2010 Mathematics Subject Classification: 30C45. Received: 30 April, 2014. Revised: 20 May, 2014. Accepted: 29 June, 2014. 267

MEANS 268 where f 1 w) = w a 2 w 2 + 2a 2 2 a 3 )w 3 5a 3 2 5a 2 a 3 + a 4 )w 4 +.... 2) We denote by S the class of analytic functions which are starlike in U. Let f A α, β R. We define the following function: F = f 1 α zf ) α] 1 β 1 α)f + αzf ] β, α, β R. 3) Remark 1. It is easy to observe that for specific values of β, the function F reduces to some generalised means. If β = 0 we obtain generalised geometric means, if β = 1 we obtain generalised arithmetic means if β = 1 we obtain generalised harmonic means of functions f zf. Definition 1. A function f A is said to be in the class H α,β, α, β R, if the function F defined by 3) is starlike, that is R { 1 β) 1 α) zf + α 1 + zf )] + β zf } + αz 2 f > 0, z U. 4) f f 1 α)f + αzf In order to prove our main results we will need the following lemmas. Lemma 1. 9, p.24] Let q Q, with q0) = a, let p = a + a n z n + be analytic in U with p a n 1. If p is not subordinate to q then there exist z 0 = r 0 e iθ0 U ζ 0 U \ Eq) m n 1 for which pu r0 ) qu) : 1. pz 0 ) = qζ 0 ), 2. z 0 p z 0 ) = mζ 0 q ζ 0 ), 3. R z 0p z 0 ) p z 0 ) + 1 mr ζ0 q ζ 0 ) q ζ 0 ) ] + 1. Denote by P the class of analytic functions p normalized by p0) = 1 having positive real part in U. Lemma 2. 6] Let p P be of the form p = 1 + p 1 z + p 2 z 2 +..., z U. Then the following estimates hold p n 2, n = 1, 2,.... Lemma 3. 4] If p P is of the form p = 1 + p 1 z + p 2 z 2 +..., z U. Then 4v + 2, v 0, p2 vp 2 1 2, 0 v 1, 4v 2, v 1.

MEANS 269 When v < 0 or v > 1, the equality holds if only if p 1 = 1 + z or one of 1 z its rotations. If 0 < v < 1 then the equality holds if only if p 1 = 1 + z2 1 z 2 or one of its rotations. If v = 0, the equality holds if only if 1 p 1 = 2 + 1 ) 1 + z 1 2 λ 1 z + 2 1 ) 1 z 2 λ, λ 0, 1] 1 + z or one of its rotations. If v = 1, the equality holds if only if p 1 is the reciprocal of one of the functions such that the equality holds in the case of v = 0. 2 Inclusion result In this section we show that the new class H α,β is a subclass of the class of starlike functions. Theorem 1. Let α, β R such that αβ1 α) 0. Then H α,β S S. Proof. Let f be in the class H α,β let p = zf f. Then from 4) we obtain that f H α,β if only if { R α1 β) zp p + αβ zp } > 0. 5) 1 α + αp Let q = 1 + z 1 z = 1 + q 1z +. 6) Then = qd) = {w : Rw > 0}, q0) = 1, Eq) = {1} q Q. To prove that f S it is enough to show that { R α1 β) zp p + αβ zp } > 0 p q. 1 α + αp Suppose that p q.then, from Lemma 1,there exist a point z 0 U a point ζ 0 U \ {1} such that pz 0 ) = qζ 0 ) Rp > 0 for all z U z0. This implies that Rpz 0 ) = 0, therefore we can choose pz 0 ) of the form pz 0 ) := ix, where x is a real number. Due to symmetry, it is sufficient to consider only the case where x > 0. We have ζ 0 = q 1 pz 0 )) = pz 0) 1 pz 0 ) + 1,

MEANS 270 then z 0 p z 0 ) = mζ 0 q ζ 0 ) = mx 2 + 1) := y, where y < 0. Thus, we obtain: R α1 β) z 0p ] z 0 ) z 0 p ] z 0 ) + R αβ = pz 0 ) 1 α + αpz 0 ) = R α1 β) y ] ] αβy y αβ1 α)] + R = 0 + ix 1 α + αix 1 α + αix 2 0. This contradicts the hypothesis of the theorem, therefore p q the proof of Theorem 1 is complete. 3 Fekete-Szegö problem In 1933 M. Fekete G. Szegö obtained sharp upper bounds for a 3 µa 2 2 for f S µ real number. For this reason, the determination of sharp upper bounds for the non-linear functional a 3 µa 2 2 for any compact family F of functions f A is popularly known as the Fekete-Szegö problem for F. For different subclasses of S, the Fekete-Szegö problem has been investigated by many authors see 2], 4], 11]). In this section we will solve the Fekete-Szegö problem for the class H α,β, where α β are positive real numbers. Theorem 2. Let α, β, µ be positive real numbers. If the function f given by 1) belongs to the class H α,β, then 4µ 2α 1)1 + αβ) + α + 1)5 + α) +, µ σ 1 + α) 2 1, 1 + 2α)1 + α) 2 a 3 µa 2 1 2, σ 1 µ σ 2, 1 + 2α where 4µ 2α 1)1 + αβ) + α + 1)3 α), µ σ 1 + α) 2 2. 1 + 2α)1 + α) 2 σ 1 = 1 + 3α αβ + α2 β, σ 2 = 2 + 5α + α2 αβ + α 2 β 21 + 2α) 21 + 2α) Proof. Let f be in the class H α, β let p P. From 4) we obtain { 1 β) 1 α) zf f + α 1 + zf )] + β zf } + αz 2 f = p. f 1 α)f + αzf Since f has the Taylor series expansion 1) p = 1+p 1 z+p 2 z 2 +..., z U, we have 1 + 1 + α)a 2 z + 21 + 2α)a 3 1 + 3α αβ + α 2 β)a 2 2] z 2 +... = 7) = 1 + p 1 z + p 2 z 2 +....

MEANS 271 Therefore, equating the coefficients of z 2 z 3 in 7), we obtain a 2 = p 1 1 + α, a 1 3 = p 2 + 1 + 3α αβ + α2 β)p 2 ] 1 21 + 2α 1 + α) 2. So, we have where v = a 3 µa 2 2 = 1 p2 vp 2 ) 1, 21 + 2α) 21 + 2α) 1 + α) 2 µ 1 + 3α αβ + α2 β 1 + α) 2. 8) Now, our result follows as an application of Lemma 3. 4 Subclass of bi-univalent function A function f A is said to be bi-univalent in U if both f f 1 are univalent in U. Let σ be the class of all functions f S such that the inverse function f 1 has an univalent analytic continuation to { w < 1}. The class σ, called the class of bi-univalent functions, was introduced by Levin 7] who showed that a 2 < 1.51. Branan Clunie 3] conjectured that a 2 2. On the other h, Netanyahu 10] showed that max a 2 = 4. Several authors have f σ 3 studied similar problems in this direction see 1] 5], 8], 12], 13]). We notice that the class σ is not empty. For example, the following functions are members of σ: z, z 1 z, log1 z), 1 2 log 1 + z 1 z. However, the Koebe function is not a member of σ. Other examples of univalent functions that are not in the class σ are z z2 2, z 1 z 2. In the sequel we assume that ϕ is an analytic function with positive real part in the unit disk U, satisfying ϕ0) = 1, ϕ 0) > 0 such that ϕu) is symmetric with respect to the real axis. Assume also that: ϕ = 1 + B 1 z + B 2 z 2 +..., B 1 > 0. 9)

MEANS 272 Definition 2. A function f A is said to be in the class H α,β ϕ), α 0, 1], β 0, if f σ satisfies the following conditions: 1 β) 1 α) zf f + α 1 + zf )] + β zf + αz 2 f f 1 α)f + αzf ϕ, 1 β) )] 1 α) wg w) + α 1 + wg w) + β wg w) + αw 2 g w) gw) g w) 1 α)gw) + αwg w) ϕw), where g is the extension of f 1 to U. Theorem 3. If f H α,β ϕ) is in A then a 2 τ B 1 B1 1 + α) + αβ1 α)]b 2 1 B 2 B 1 )1 + α) 2, 10) ] 1 a 3 B 1 1 + α + αβ1 α) + B 2 B 1 21 + 2α)1 + α) 1 + α. 11) Proof. Let f H α,β ϕ) g = f 1. Then there exist two analytic functions u, v : U U with u0) = v0) = 0 such that: 1 β) 1 α) zf + α 1 + zf )] + β zf + αz 2 f = ϕu) f f 1 α)f + αzf )] 1 β) 1 α) wg w) + α 1 + wg w) + β w w) + αw 2 g w) gw) g w) 1 α)gw) + αwg w) = ϕvw)). 12) Define the functions p q by p = 1 + u 1 u = 1 + p1z + p2z2 +..., q = 1 + v 1 v = 1 + q1z + q2z2 +... or equivalently, u = p 1 p + 1 = 1 2 ) ] p 1 z + p 2 p2 1 2 z2 +..., 13) v = q 1 q + 1 = 1 ) ] q 1 z + q 2 q2 1 2 2 z2 +.... 14)

MEANS 273 We observe that p, q P, in view of Lemma 2, we have that p n 2 q n 2, for n 1. Further, using 13) 14) together with 9), it is evident that ϕu) = 1 + 1 2 B 1p 1 z + 1 2 B 1 p 2 1 2 p2 1 ) + 1 ] 4 B 2p 2 1 z 2 +... 15) ϕv) = 1 + 1 1 2 B 1q 1 z + 2 B 1 q 2 1 ) 2 q2 1 + 1 ] 4 B 2q1 2 z 2 +.... 16) Therefore, in view of 12), 15) 16) we have 1 β) 1 α) zf f + α 1 + zf )] f + β zf + αz 2 f 1 α)f + αzf 1 β) = 1 + 1 1 2 B 1p 1 z + 2 B 1 p 2 1 ) 2 p2 1 + 1 ) 4 B 2p 2 1 z 2 +..., 17) )] 1 α) wg w) + α 1 + wg w) gw) g + β wg w) + αw 2 g w) w) 1 α)gw) + αwg w) = 1 + 1 1 2 B 1q 1 w + 2 B 1 q 2 1 ) 2 q2 1 + 1 ) 4 B 2q1 2 w 2 +.... 18) Since f σ has the Taylor series expansion 1) g = f 1 the series expansion 2), we have 1 β) 1 α) zf f + α 1 + zf )] f + β zf + αz 2 f 1 α)f + αzf = 1 + 1 + α)a 2 z + 21 + 2α)a 3 1 + 3α αβ + α 2 β)a 2 2] z 2 +..., 19) 1 β) )] 1 α) wg w) + α 1 + wg w) gw) g + β wg w) + αw 2 g w) w) 1 α)gw) + αwg w) = 1 1 + α)a 2w 21 + 2α)a 3 1 + 3α αβ + α 2 β)a 3 2a 2 2) ] w 2 +.... 20)

MEANS 274 Equating the coefficients in 17), 19) 18), 20), we obtain 1 + α)a 2 = 1 2 B1p1, 21 + 2α)a 3 1 + 3α αβ + α 2 β)a 2 2 = 1 ) 2 B1 p 2 p2 1 + 1 2 4 B2p2 1, 1 + α)a 2 = 1 2 B1q1, 21 + 2α)a 3 2a 2 2) 1 + 3α αβ + α 2 β)a 2 2 = 1 ) 2 B1 q 2 q2 1 + 1 2 4 B2q2 1. 21) From the first the third equation of the system 21) it follows that a 2 2 = p 1 = q 1, 22) ) 2 B1 p 1 τ. 23) 41 + γ) Now, 22), 23) the next two equations of the system 21) lead to a 2 2 = B 3 1p 2 + q 2 ) 41 + α) + αβ1 α)]b 2 1 4B 2 B 1 )1 + α) 2. 24) Thus, in view of Lemma 2, we obtain the desired estimation of a 2. From the third the fourth equation of 21), we obtain a 3 = 1 2 B 1p 2 3 + 5α + αβ1 α) 41 + 2α)1 + α) + 1 + 3α αβ1 α) 41 + 2α)1 + α) + 1 1 4 p2 1B 2 B 1 ) 1 + α, which yields to the estimate given by 11) so the proof of Theorem 3 is completed. References 1] R. M. Ali, S. K. Lee, V. Ravichran, S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike convex functions, Appl. Math. Lett. 252012), 344-351. 2] D. Bansal, Fekete-Szegö problem for a new class of analytic functions, Int. J. Math. Mathematical Sciences 20112011), article ID: 143096, 5 pages.

MEANS 275 3] D. A. Branan, J. G. Clunie, W. E. Kirwan, Coefficient estimates for a class of starlike functions, Canad. J. Math. 221970), 476-485. 4] M. Fekete, G. Szegö, Eine Bemerkung über ungerade schlichte Functionen, J. London Math. Soc. 81933), 85-89 5] S. P. Goyal, P. Goswami, Estimate for initial Maclaurin coefficients of biunivalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc. 202012), issue 3,179-182. 6] A. W. Goodman, Univalent functions, Polygonal Publishing House, Washington, New Jersey, 1983. 7] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 181967), 63-68. 8] X.-F. Li, A.-P. Wang, Two new subclasses of bi-univalent functions, Int. Math. Forum no. 30, 72012), 1495-1504. 9] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory Applications, Dekker, New York, 2000. 10] E. Netanyahu, The minimal distance of the image boundary from the origin the second coefficient of a univalent function in z < 1, Arch. Ration. Mech. Anal. 321969), 100-112. 11] H. M. Srivastava, A. K. Mishra, M. K. Das, The Fekete-Szegö problem for a subclass of close-to-convex functions, Complex Variables, Theory Appl. 442001), 145-163. 12] H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic bi-univalent functions, Appl. Math. Lett. 232010), 1188-1192. 13] Q.-H. Xu, Y.-C. Gui, H.M. Srivastava, Coefficient estimates for a certain subclass of analytic bi-univalent functions, Appl. Math. Lett. 252012), 990-994. Andreea-Elena TUDOR, Department of Mathematics, Transilvania University of Braşov, Str.Iuliu Maniu 50, 500091, Brasov, Romania. Email: tudor reea elena@yahoo.com Dorina RĂDUCANU, Department of Mathematics, Transilvania University of Braşov, Str.Iuliu Maniu 50, 500091, Brasov, Romania. Email: draducanu@unitbv.ro

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