PROPERTIES AND CHARACTERISTICS OF A FAMILY CONSISTING OF BAZILEVIĆ (TYPE) FUNCTIONS SPECIFIED BY CERTAIN LINEAR OPERATORS

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1 Electronic Journal of Mathematical Analysis and Applications Vol. 7) July 019, pp ISSN: X online) PROPERTIES AND CHARACTERISTICS OF A FAMILY CONSISTING OF BAZILEVIĆ TYPE) FUNCTIONS SPECIFIED BY CERTAIN LINEAR OPERATORS A.R.S. JUMA, S.N. AL-KHAFAJI, H. IRMAK Abstract. In this paper, through the instrument of the well-nown Gauss hypergeometric function and Hadamard product, a linear operator is firstly defined and a family consisting of Bazilević type) function is then described by using the related operator. Several systematic investigations of the various properties and characteristics of the concerned family are also presented. 1. Introduction, Definitions and Preliminaries Let R + = 0, ) be the set of positive real numbers, N = { 1,, 3, } be the set of positive integers, Z = { 1,, 3, } be the set of negative integers, C be the set of complex numbers and also let N 0 = N { 0}, Z 0 = Z { 0} and R + 0 = R+ { 0}. We let A denote the family of functions fz) which are analytic in the open unit dis U = { z : z C and z < 1 } and normalized with the following conditions f0) = 0 and f 0) = 1. Clearly, these functions fz) have the following Taylor-Mclaurin series form: fz) = z + a z + a 3 z a z + a 0; z U ). 1) In the light of well-nown binomial expansion, we can now introduce a novel operator denoted by D n,λ [f] or Dn,λ [f]z) also defined by D n,λ [f] := zα + c z +α 1, ) 010 Mathematics Subject Classification. 30A10, 30C45, 30C50, 30C55; 6D10. Key words and phrases. Linear operator, Hadamard Product, Maximum modulus theorem, Complex plane, Open unit dis, Inequalities in the complex plane, Analytic function, Bazilević type) function, Extremal function. Submitted Aug. 13,

2 40 A.R.S. JUMA, S.N. AL-KHAFAJI, H. IRMAK EJMAA-019/7) where and ) + λ 1) c := c n, l, α, λ a := ) na f fz) A, n N 0, λ R + 0, α R+, and z U. As a natural consequence of the operator given by ), it can be easily determined by the following relationships: D 0,λ [f] = zα + a z +α 1 3) and also D 1,λ l λα + α ) ) [f] = D 0,λ [f] + D 0,λ λz [f] + λ 1) ) = z α + a z +α 1 4) D,λ [f] = zα + + λ 1) ) a z +α 1, 5) where f fz) A, λ R + 0, α R+ and z U. For our main results, we need also to recall the Gauss hypergeometric function defined by a) b) F 1 a, b, c; z) = z z U), 6) =0 where a, b C, c Z 0 and ) n denotes Pochhammer symbol or the shifted factorial) given, in terms of the Gamma function Γ, by { Γ + n) + 1) + ) + n 1) if n N ) n = 7) Γ) 1 if n = 0. Next, corresponding to the function F 1 a, b, c; z), we introduce a linear operator: H n,λ [f]z) or Hn,λ [f] which is defined by means of the following Hadamard product or convolution): of the functions or, equaivalently, by D n,λ [f]z) F 1 a, b, c; z) D n,λ [f]z) and F 1 a, b, c; z) H n,λ [f] := Dn,λ [f]z) F 1 a, b, c; z) + λ 1) ) n = z α a) b) + a z +α 1, 8) where n N 0, λ R + 0, α R+, c Z 0, fz) A and z U. Finally, under the conditions of the definitions in 1), 6), and 7) and also by maing use of the above-linear operator H n,λ [f], we say that a function f := fz)

3 EJMAA-019/7) PROPERTIES AND CHARACTERISTICS OF CERTAIN FUNCTIONS 41 A, given by 1), is in the aforementioned family B n,λ β, γ, δ), which also consists of Bazilević type) functions, if it satisfies the following inequality: ) z 1+α H n,λ [f] αz α ) ) δ 1 z 1+αH n,λ [f] ) < β z U ) 9) δγ αz α for all n N 0, λ R + 0, α R+, 0 < β 1, 0 γ 1 and 0 δ 1. In the literature, from time to time, we encounter several wors associated with a great number of operators linear or nonlinear) constituted by certain complex functions with one variable or several variables) which are analytic or univalent in certain domains of the complex plane C. As examples, one can see those given by the papers in [3], [4], [7], [8], [9], [10], [11], [1], and also [13]. Moreover, since some of them are very important for the theory of functions, intrinsically, they or some of them have also an important roles for analytic or geometric function theory, which is a special field of complex analysis and includes the study of the relations between the analytic properties of a function fz) and the geometric properties of the image domain fu). For these properties of certain complex functions, one may also refer to the wors in [1], [], [5], [6] and also others. By virtue of the reasons indicated by above, a function fz) A in the family 9) defined by the operator in 8) is of significance for its analytic and geometric properties. For this, firstly, the authors introduced a new and also general family B n,λ β, γ, δ) of functions which are Bazilević type-univalent) functions in the open unit dis U and then obtained a necessary and sufficient condition for a function to be in the family B n,λ β, γ, δ). They also presented several basic results, derived here for functions in the family B n,λ β, γ, δ), in relation with some growth and distortion theorems, the radii of the Bazilević type-univalent) functions and certain extremal functions. In addition, they also pointed some interesting and possible other results of the basic results out.. Basic Properties and Characteristics of the Family B n,λ β, γ, δ) and Some of its Consequences We begin by giving and then proving a characterization property of the abovedefined family B n,λ β, γ, δ), which is contained in the following theorem. Theorem 1. Let the function fz) B n,λ β, γ, δ) be in the form 1). Then, fz) is in the family B n,λ β, γ, δ) if and only if + λ 1) + α 1)a 1 βδ + β, 10) where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, and n N 0. The result is sharp for the extremal function given by c) 1) + λ ) nz fz) = z +, 11) 1 + α)1 βδ + β)a) b)

4 4 A.R.S. JUMA, S.N. AL-KHAFAJI, H. IRMAK EJMAA-019/7) where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, n N 0, and z U Proof. Firstly, assume that the function fz) B n,λ β, γ, δ) is defined by 1) and the inequality 10) holds true. Then, for z U), we calculate that z α+1 ) H n,λ [f] αz α βδ 1)z α+1 ) H n,λ [f] δγ αz α ) + λ 1) = + α 1)a z α+ 1 β [ + λ 1) ) n δ 1) αz α a) b) + + λ 1) + α 1)a z α+ 1] δγ αz α) + α 1)a 1 βδ + β) 0, by virtue of the desired inequality in 10). Hence, by maximum modulus theorem, the function fz) belongs to the family B n,λ β, γ, δ). To prove the converse, suppose that the function fz) is defined by 1) and is in the family B n,λ β, γ, δ). So that the condition 9) readily yields that ) z α+1 H n,λ [f] αz α ) δ 1)z α+1 H n,λ [f] δγ αz α ) z α+1 [αz α 1 + = l+α+λ 1) l+α ) n a) b) α + 1)a z α+ ] αz α δ 1)z α+1 [αz α 1 + l+α+λ 1) l+α ) n a) b) α + 1)a z α+ ] δγ + αz α < β. 1) Since Rez) z for any z U, if we choose z to be real and let z 1, we shall find from 1) that + λ 1) α + 1)a α) + λ 1) β δ 1) which readily yields that the desired assertion 10). α + 1)a α) + δα γ) Finally, by observing that the function fz) defined by 1) is indeed an extremal function for the assertion 10), the proof of Theorem 1 is therefore completed. The next theorem is also contained in the following form. Theorem. The family B n,λ β, γ, δ) is closed under convex linear combination. Proof. Let the functions f j z) = z + a,j z a,j 0; j = 1, ), 13) ),

5 EJMAA-019/7) PROPERTIES AND CHARACTERISTICS OF CERTAIN FUNCTIONS 43 be in the family B n,λ β, γ, δ). Then, we have to show that the function hz) defined by ) hz) = λf 1 z) + 1 λ)f z) 0 λ 1, 14) is in the family B n,λ β, γ, δ). If one taes cognizance of the assertion 10) for the definition of functions f j z) j=1,) defined by 13) and also for the function hz) defined by 14), or, equivalently, in the form: [ hz) = z + λa,1 z + 1 λ)a, z ], the inequality: { + λ 1) + α 1) [ λa,1 α) + 1 λ)a, α) ]} 1 βδ + β), is easily obtained. Thereby, that function hz) belongs to B n,λ β, γ, δ) and the desired proof is here completed. The following result is more generalization of Theorem and it can be similarly proven, which is contained in the following theorem. Theorem 3. Let each of the functions f i z) be defined by f j z) = z + a 1,j z + a,j z + + a,j z + and also be in the same family B n,λ β, γ, δ), where Then, the function hz) defined by where a, j 0 ; j = 1,,, m ; m N and z U. hz) = c 1 f 1 z) + c f z) + + c m f m z), c j 0 j = 1,,, m ) and c 1 + c + + c m = 1, is also in the family B n,λ β, γ, δ). The growth and distortion theorems for the function fz), defined by 1), in the family B n,λ β, γ, δ) are also presented by the following theorems, which are Theorems 4 and 5 below. Theorem 4. Let the function fz) be in the family B n,λ β, γ, δ). Then, the following inequalities: fz) c) 1) + λ ) n z z + 15) 1 + α)1 βδ + β)a) b) and fz) c) 1) + λ ) n z z 1 + α)1 βδ + β)a) b) 16)

6 44 A.R.S. JUMA, S.N. AL-KHAFAJI, H. IRMAK EJMAA-019/7) are satisfied, where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, and n N 0 and z U. The bounds in 15) and 16) are attained for the function fz) given by fz) = z + c) 1) + λ 1 + α)1 βδ + β)a) b) ) nz z U). 17) Proof. Let fz), defined by 1), be in the family B n,λ β, γ, δ). By using 10), we then obtain + λ ) n a) b) 1 + α) a c) 1) + λ 1) + α 1)a 1 βδ + β, which yields c) 1) + λ ) n n ) a N α)1 βδ + β)a) b) By the help of the inequality just above, we get that fz) = z + a z z + z z + α l+α+λ l+α and fz) = z a z z z a a )n a)b) c) 1) 1 βδ + β) z z α l+α+λ a)b) l+α )n c) 1) 1 βδ + β) z, where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, and n N 0 and z U. Thus, the desired prof completes. Theorem 5. Let the function fz) be in the family B n,λ β, γ, δ). Then, the following inequalities: f z) 1 + and c) 1) + λ ) n z 18) 1 + α)1 βδ + β)a) b) f z) c) 1) + λ ) n z 1 19) 1 + α)1 βδ + β)a) b) are also satisfied, where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, and n N 0 and z U. The bounds in 18) and 19) are attained for the function fz) given by 17).

7 EJMAA-019/7) PROPERTIES AND CHARACTERISTICS OF CERTAIN FUNCTIONS 45 Proof. Let fz), defined by 1), be in the family B n,λ β, γ, δ). By using 10), we then obtain ) n a) b) 1 + α + λ c) 1) = a + λ 1) + α 1)a + λ 1) + α 1)a 1 βδ + β, which yields c) 1) + λ ) n a 1 + α)1 βδ + β)a) b) n ) N0. 0) By means of the inequality given by 0), the assertions in 19) and 0) can be easily obtained. The details are here omitted. The last result in regard to the radii of Bazilević type-univalent) function of order ω 0 ω < 1) is contained in the following theorem. Theorem 6. Let fz), defined by 1), be in the family B n,λ β, γ, δ). Then, fz) is Bazilevic univalent) function of order ω 0 ω < 1) in dis z < r, where r := rn, l, λ, n, α, β, γ, δ, ω) 1 ω)a) b) + α 1)1 βδ + β) + λ 1) ) ) 1 n 1 = inf, 1) ω ) where a > 0, b > 0, c > 0, λ > 0, α > 0, 0 < β 1, 0 γ 1, 0 δ 1, α γ 0, and n N 0. The result is sharp for the extremal function given by fz) = z + for all = 3, 4, and z U. c) 1) + λ 1) + α 1)1 βδ + β)a) b) ) nz Proof. Let fz) B n,λ β, γ, δ) be in the form??). Then, the function fz) is Bazilević type-univalent) function of order ω 0 ω < 1) in z < r, provided that [ zf z) ] f 1 z) = 1)a z a z 1 1)a z a z 1 1 ω z < r; 0 ω < 1 ), 3) where r is given by 1). As is nown, the last inequality in 3) holds true if ω ) z 1 1 βδ + β)α + 1)a) b) + λ 1) 1 ω ) n, )

8 46 A.R.S. JUMA, S.N. AL-KHAFAJI, H. IRMAK EJMAA-019/7) which yields the expression in 1), when it is solved for z. This completes the proof of Theorem 6. It is easy to see that all theorems, which are Theorems 1-6, include several special or general results. For all of them, it is enough to choose the values of the parameters which are available in all theorems. As certain consequences of our results, we want to present only four. Under the conditions of Theorem 1, an immediate consequence of Theorem 1 may be given by the following corollary, which is Corollary 1. Corollary 1. If the function fz) given by 1) is in B n,λ a for all =, 3,. β, γ, δ), then c) 1) + λ 1) + α 1)1 βδ + β)a) b) Under the conditions of Theorem 1 and as a special application of the operator 1), by setting n := 0 in Theorem 1, the next consequence can be presented by the following corollary. Corollary. If the function fz) given by 1) is in B 0,λ β, γ, δ) if and only if a) b) + α 1)a The result is sharp for the extremal function given by fz) = z + 1 βδ + β. c) 1) 1 + α)1 βδ + β)a) b) z z U ). Under the conditions of Theorem 4 and as a special application of the operator 5), by setting n := 1 in Theorem 4, the third consequence can be also given by the following corollary. Corollary 3. If the function fz) given by 1) is in B 1,λ fz) z ) n β, γ, δ), then )c) 1) 1 + α)1 βδ + β) + λ)a) b) z z U. ) 4) The above bounds are attained for the function fz) given by fz) = z + )c) 1) 1 + α)1 βδ + β) + λ)a) b) z z U). 5) Under the conditions of Theorem 6 and as a special application of the operator 5), by letting n := in Theorem 6, the last consequence of our results can be given by the following corollary. Corollary 4. If the function fz) given by 1) is in B 1,λ β, γ, δ), then fz) is Bazilevic type-univalent) function of order ω 0 ω < 1) in dis z < r, where r := r l, λ, n, α, β, γ, δ, ω) = inf 1 ω)a) b) + α 1)1 βδ + β) + λ 1) ) ) 1 1. ω )

9 EJMAA-019/7) PROPERTIES AND CHARACTERISTICS OF CERTAIN FUNCTIONS 47 The result is sharp for the extremal function given by c) 1) ) z fz) = z + + α 1)1 βδ + β)a) b) + λ 1) for all = 3, 4, and z U. References [1] R.M. Aliy, M.H. Mohdz, L.S. Keongx, Radii of starlieness, parabolic starlieness and strong starlieness for Janowsi starlie functions with complex parameters, Tamsui Oxford J. Infor. Math. Sci., 7, 3, 53-67, 011. [] M.K. Aouf, R.M. El-Ashwah and S.M. El-Deebm, Certain classes of univalent functions with negative coefficients and n-starlie with respect to certain points, Mat. Vesni, 6, 3, 15-6, 010. [3] M.P. Chen, H. Irma and H.M. Srivastava, A certain subclass of analytic functions involving operators of fractional calculus, 35, 5, 83-91, [4] H.E. Darwish, A.Y. Lashin, E.M. Madar, On certain classes of univalent functions with negative coefficients defined by convolution. Electron. J. Math. Anal. Appl., 4, 1,, , 016. [5] P.L. Duren, Univalent Functions, Grundlehren der Mathematishen Wissenschaften 59, Springer-Verlag, [6] A.W. Goodman, Univalent Functions. Vols. I and II, Polygonal Publishing Company, [7] H. Irma, The fractional differ-integral operators and some of their applications to certain multivalent functions. J. Fract. Calc. Appl., 8, 1, , 017. [8] H. Irma, Certain complex equations created by integral operator and some of their applications to analytic functions, Le Math., 71, 1, 43-49, 016. [9] H. Irma, The ordinary differential operator and some of its applications to p-valently analytic functions, Electron. J. Math. Anal. Appl., 4, 1, 05-10, 016. [10] H. Irma and N.E. Cho, A differential operator and its applications to certain multivalently analytic functions, Hacet. J. Math. Stat., 36, 1, 1-6, 007. [11] H. Irma, S.H. Lee, N.E. Cho, Some multivalently starlie functions with negative coefficients and their subclasses defined by using a differential operator, Kyungpoo Math. J., 37, 1, 43-51, [1] R.A.S. Juma, S.R. Kularni, On univalent functions with negative coefficient by using generlized Salagean operator, Fillomat, 1,, , 007. [13] M. Şan and H. Irma, Ordinary differential operator and some of its applications to certain meromorphically p-valent functions, Appl. Math. Comput., 18, 3, , 011. Abdul Rahman S. Juma Department of Mathematics, University of Anbar, Ramadi, IRAQ address: dr juma@hotmail.com or eps.abdulrahman.juma@uoanbar.edu.iq Saba Nazar Al-hafaji Department of Mathematics, AL-Mustansiriyah University, IRAQ address: sabanf.mc11p@uoufa.edu.iq Hüseyin Irma Department of Mathematics, Faculty of Sciences, Çanırı Karatein University, Tr 18100, Çanırı, TURKEY address: hisimya@yahoo.com or hirma@aratein.edu.tr