On second-order differential subordinations for a class of analytic functions defined by convolution

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1 Available online at J. Nonlinear Sci. Appl., 1 217), Research Article Journal Hoepage: On second-order differential subordinations for a class of analytic functions defined by convolution Arzu Akgül Kocaeli University, Faculty of Arts and Sciences, Departent of Matheatics, Uuttepe Capus, İzit-Kocaeli, Turkey. Counicated by E. Savaş Abstract Making use of the convolution operator we introduce a new class of analytic functions in the open unit disk and investigate soe subordination results. c 217 All rights reserved. Keywords: Analytic functions, univalent function, differential subordination, convex function, Hadaard product, best doinant. 21 MSC: 3C Introduction Let C be coplex plane and U = z C : z < 1} = U\ }, open unit disc in C. Let HU) be the class of analytic functions in U. For p N + = 1, 2, 3, } and a C, let H[a, k] be the subclass of HU) consisting of the functions of the for fz) = a + a k z k + a k+1 z k+1 + with H H[, 1] and H H[1, 1]. Let A p be the class of all analytic functions of the for fz) = z p + a k z k 1.1) k=p+1 in the open unit disk U with A 1 = A. For functions f A p given by equation 1.1) and g A p defined by gz) = z p + b k z k, k=p+1 their Hadaard product or convolution) [7] of f and g is defined by := z p + a k b k z k. k=p+1 A function f HU) is univalent if it is one to one in U. Let S denote the subclass of A consisting of Eail address: akgul@kocaeli.edu.tr Arzu Akgül) doi: /jnsa Received

2 A. Akgül, J. Nonlinear Sci. Appl., 1 217), functions univalent in U. If a function f A aps U onto a convex doain and f is univalent, then f is called a convex function. Let } } K = f A : R 1 + zf z) >, z U f z) denote the class of all convex functions defined in U and noralized by f) =, f ) = 1. Let f and F be ebers of HU). The function f is said to be subordinate to ϕ, if there exists a Schwarz function w analytic in U with w) = and wz) < 1, z U), such that We denote this subordination by fz) = ϕwz)). fz) ϕz) or f ϕ. Furtherore, if the function ϕ is univalent in U, then we have the following equivalence [5, 13] fz) ϕz) f) = ϕ) and fu) ϕu). The ethod of differential subordinations also known as the adissible functions ethod) was first introduced by Miller and Mocanu in 1978 [11] and the theory started to develop in 1981 [12]. All the details were captured in a book by Miller and Mocanu in 2 [13]. Let Ψ : C 3 U C and h be univalent in U. If p is analytic in U and satisfies the second-order differential subordination ) Ψ pz), zp z), zp z); z hz), 1.2) then p is called a solution of the differential subordination. The univalent function q is called a doinant of the solution of the differential subordination or ore siply doinant, if p q for all p satisfying 1.2). A doinant q 1 satisfying q 1 q for all doinants 1.2) is said to be the best doinant of 1.2). For functions f, g A p, the linear operator,p : A p A p, N }) is defined by: Thus, we get,p =,p Fro 1.3) it can be easily seen that z p,p =, 1,p =,p ) = 1 ) + z ) p = z p p + k p) + a k b k z k, p k=p+1 2,p =,p [, p]. 1,p ) = z p + k=p+1 ) p + k p) a k b k z k, ). 1.3),p ) = +1,p 1 ),p, ). The operator,p f g) was introduced and studied by Selveraj and Selvakuaran [19], Aouf and Mostafa [4], and for p = 1 was introduced by Aouf and Mostafa [3]. Recent years, Özkan [16], Özkan p

3 A. Akgül, J. Nonlinear Sci. Appl., 1 217), and Altntaş [17], Lupaş [9], and Lupaş [1] also see [1, 2]) investigated soe applications and results of subordinations of analytic functions given by convolution. Also Bulut [6] used the sae techniques by using Koatu integral operator. In soe of this study, the results given by Lupaş [1] and Lupaş [9] were generalized. In order to prove our ain results we need the following leas. Lea 1.1 [8]). Let h be convex function with h) = a and let γ C := C\} be a coplex nuber with Rγ}. If p H[a, k] and then where qz) = pz) + 1 γ zp z) hz), 1.4) pz) qz) hz), γ nz γ/n z t γ/n) 1 ht)dt. The function q is convex and is the best doinant of the subordination 1.4). Lea 1.2 [15]). Let Rµ} >, n N, and let w = n2 + µ 2 n 2 µ 2. 4nRµ} Let h be an analytic function in U with h) = 1 and suppose that } R 1 + zh z) > w. h z) If is analytic in U and pz) = 1 + p n z n + p n+1 z n+1 + pz) + 1 µ zp z) hz), 1.5) then where q is a solution of the differential equation pz) qz), qz) + n µ zq z) = hz), q) = 1, given by qz) = µ nz σ/n z t µ/n ) 1 ht)dt Moreover q is the best doinant of the subordination 1.5). z U). Lea 1.3 [14]). Let r be a convex function in U and let hz) = rz) + nβzr z), z U), where β > and n N. If pz) = r) + p n z n + p n+1 z n+1 +, z U)

4 A. Akgül, J. Nonlinear Sci. Appl., 1 217), is holoorphic in U and then and this result is sharp. pz) + βzp z) hz), pz) rz), In the present paper, aking use of the subordination results of [13] and [18] we will prove our ain results. Definition 1.4. Let R, β) be the class of functions f A satisfying R ) } > β, where z U, β < Main results Theore 2.1. The set R, β) is convex. Proof. Let ) fj g j z) = z + a k,j b k,j z k z U; j = 1,...) be in the class of R, β). Then, by Definition 1.4, we get R f j g j )z) ) } } = R 1 + a k,j b k,j kz k 1 > β. For any positive nubers 1, 2,..., l such that j = 1, we have to show that the function is eber of R, β), that is, hz) = ) f j g j z) R hz)) } > β. 2.1) Thus, we have hz) = z k 1)) j a k,j b k,j z k. 2.2) If we differentiate 2.2) with respect to z, then we obtain hz)) = k 1)) j a k,j b k,j kz k 1

5 A. Akgül, J. Nonlinear Sci. Appl., 1 217), = 1 + j 1 + k 1)) a k,j b k,j kz k 1. Thus, we have R hz)) } = 1 + > 1 + = β. } j R 1 + k 1)) a k,j b k,j kz k 1 j β 1) Thus, the inequality 2.1) holds and we obtain desired result. Theore 2.2. Let q be convex function in U with q) = 1 and let hz) = qz) + 1 γ + 1 zq z) z U), where γ is a coplex nuber with Rγ} > 1. If f R σ,θ β) and F = Υ γ f g), where Fz) = Υ γ = γ + 1 z γ z t γ 1 f g)t)dt, 2.3) then, iplies and this result is sharp. ) hz) 2.4) Fz)) qz), Proof. Fro the equality 2.3) we can write z z γ Fz) = γ + 1) t γ 1 f g)t)dt, 2.5) by differentiating the equality 2.5) with respect to z, we obtain γ) Fz) + zf z) = γ + 1). If we apply the operator to the last equation, then we get If we differentiate 2.6) with respect to z, we can obtain γ) Fz) + z Fz)) = γ + 1). 2.6) Fz)) + 1 γ + 1 z Fz)) = fz)). 2.7) By using the differential subordination given by 2.4) in the equality 2.7), we have Fz)) + 1 γ + 1 z Fz)) hz). 2.8)

6 A. Akgül, J. Nonlinear Sci. Appl., 1 217), Now, we define Then by a siple coputation we get pz) = [ z + ] 1 + k 1)) γ + 1 γ + k a kb k z k Using the equation 2.9) in the subordination 2.8), we obtain If we use Lea 1.2, then we get pz) = Fz)). 2.9) pz) + 1 γ + 1 zp z) hz) = qz) + 1 γ + 1 zq z). pz) qz). So we obtain the desired result and q is the best doinant. Exaple 2.3. If we choose in Theore 2.2 γ = i + 1, qz) = 1 1 z, = 1 + p 1 z + p 2 z +, p H[1, 1]). thus we get If f g) R, β) and F is given by hz) = i + 2) i + 1)z i + 2)1 z) 2. Fz) = Υ i = i + 2 z i+1 z t i f g)t)dt, then by Theore 2.2, we obtain fz)) hz) = i + 2) i + 1)z i + 2)1 z) 2 = Fz)) qz)) = 1 1 z. Theore 2.4. Let Rγ} > 1 and let w = 1 + γ γ 2 + 2γ. 4Rγ + 1} Let h be an analytic function in U with h) = 1 and assue that } R 1 + zh z) > w. h z) If f g R, β) and F = Υ δ γf g), where F is defined by equation 2.3), then ) hz) 2.1) iplies Fz)) qz),

7 A. Akgül, J. Nonlinear Sci. Appl., 1 217), where q is the solution of the differential equation given by hz) = qz) + 1 γ + 1 zq z), q) = 1, qz) = γ + 1 z γ+1 Moreover q is the best doinant of the subordination 2.1). z t γ f g)t)dt. Proof. If we choose n = 1 and µ = γ + 1 in Lea 1.2, then the proof is obtained by eans of the proof of Theore 2.4. Letting 1 + 2β 1)z hz) =, β < z in Theore 2.4, we obtain the following result. Corollary 2.5. If β < 1, ξ < 1,, Rγ} > 1, and F = Υ γ f g) is defined by the equation 2.3), then Υ γ R, β)) R, ξ), where and this result is sharp. Also, where ξ = in Rqz)} = ξγ, β) z =1 ξ = ξγ, β) = 2β 1) + 2γ + 1)1 β)τγ), 2.11) τγ) = Proof. Let f R, β). Then fro Definition 1.4 it is known that R ) } > β, which is equivalent to By using Theore 2.2, we have If we take hz) = then h is convex and by Theore 2.4, we obtain Fz)) qz) = γ + 1 z γ+1 z 1 ) hz). Fz)) qz). t γ dt. 2.12) 1 + t 1 + 2β 1)z, β < 1, 1 + z γ 1 + 2β 1)t t 1 + t 1 β)γ + 1) z dt = 2β 1) + 2 z γ+1 t γ 1 + t dt. On the other hand if Rγ} > 1, then fro the convexity of q and the fact that qu) is syetric with respect to the real axis, we get R Fz)) } in Rqz)} = Rq1)} = ξγ, β) = 2β β)γ + 1)τγ), 2.13) z =1

8 A. Akgül, J. Nonlinear Sci. Appl., 1 217), where τγ) is given by equation 2.12). Fro inequality 2.13), we get Υ γ R, β)) R, ξ), where ξ is given by 2.11). Theore 2.6. Let q be a convex function with q) = 1 and h a function such that hz) = qz) + zq z), z U). If f A, then the following subordination ) hz) = qz) + zq z) 2.14) iplies that and the result is sharp. Proof. Let Differentiating 2.15), we have ) qz), z pz) = ). 2.15) z ) = pz) + zp z), z U) and thus 2.14) becoes pz) + zp z) hz) = qz) + zq z). Hence by applying Lea1.3, we conclude that pz) qz), that is, ) and this result is sharp. z qz), Theore 2.7. Let q be a convex function with q) = 1 and h the function hz) = qz) + zq z) z U). If N, f A and verifies the differential subordination +1 ) hz), 2.16) then and this result is sharp. +1 qz), Proof. For the function f A, given by the equation 1.1), we have = z k 1)) γ + 1 k + γ a kb k z k, z U).

9 A. Akgül, J. Nonlinear Sci. Appl., 1 217), Let us consider z + +1 γ k 1)) pz) = +1 k+γ a kb k z k = z k 1)) γ+1 k+γ a kb k z k 1 + = γ k 1)) k+γ a kb k z k k 1)) γ+1 k+γ a kb k z k 1. We get pz)) = +1 ) pz) ) ) and pz) + zp z) = z +1 ) z U). Thus, the relation 2.16) becoes pz) + zp z) hz) = qz) + zq z), z U), and by using Lea 1.3, we obtain pz) qz), that is, ) z qz). Acknowledgent The author would like to thank the anonyous referees for their valuable suggestions which iproved the presentation of the paper. References [1] Ş. Altınkaya, S. Yalçın, Coefficient estiates for two new subclasses of biunivalent functions with respect to syetric points, J. Funct. Spaces, ), 5 pages. 1 [2] Ş. Altınkaya, S. Yalçın, General properties of ultivalent concave functions involving linear operator of carlson-shaffer type, Coptes rendus de lacade ie bulgare des Sciences, ), [3] M. K. Aouf, A. O. Mostafa, Sandwich theores for analytic functions defined by convolution, Acta Univ. Apulensis Math. Infor., 21 21), [4] M. K. Aouf, A. O. Mostafa, Subordination results for a class of ultivalent non-bazilevic analytic functions defined by linear operator, Acta Univ. Apulensis Math. Infor., ), [5] T. Bulboacă, Differential subordinations and superordinations, Recent Results, Casa Cărţii de Ştiinţă, Cluj-Napoca, 25). 1 [6] S. Bulut, Soe applications of second-order differential subordination on a class of analytic functions defined by Koatu integral operator, ISRN Math. Anal., ), 5 pages. 1 [7] P. L. Duren, Univalent functions, Grundlehren der Matheatischen Wissenschaften [Fundaental Principles of Matheatical Sciences], Springer-Verlag, New York, 1983) 1 [8] D. J. Hallenbeck, S. Ruscheweyh, Subordination by convex functions, Proc. Aer. Math. Soc., ),

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