A NOTE ON RUSCHEWEYH TYPE OF INTEGRAL OPERATORS

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1 IJMMS 29: PII S Hindawi Publishing Corp A NOTE ON RUSCHEWEYH TYPE OF INTEGRAL OPERATORS FOR UNIFORMLY -CONVEX FUNCTIONS M ANBU DURAI and R PARVATHAM Received 7 March 2001 We prove that the class of uniformly -convex functions introduced by Kanas is closed under the generalized Ruscheweyh integral operator for 0 < Mathematics Subject Classification: 30C45 We denote by the class of functions fz= z + a 2 z 2 + which are analytic in =z C : z < 1 LetS denote the class of functions in that are univalent in The subclasses of S containing functions which are uniformly convex and uniformly starlike, introduced by Goodman [1, 2, are denoted by UCV and UST, respectively The class of uniformly -convex functions was introduced by Kanas [3 and she gave an analytic condition for such functions as follows: fzis a uniformly -convex function if and only if Re 1 z ζf z fz fζ + 1+ z ζf z > 0 1 for all z,ζ and 0 1 For ζ = 0, this class of functions reduces to Mocanu s class M of -convex functions [4 In this note, for >0, we consider the integral operator Fz= F z,ζ F 0,ζ F 0,ζ, 2 where c +1/ z F z,ζ = t ζ c 1 ft fζ 1/ dt 3 z ζ c ζ for all z and for fixed ζ with z ζ We prove that this normalized function Fz is a uniformly -convex function when fzis a uniformly -convex function in the sense of Kanas [3 For ζ = 0 the operator Fz reduces to Ruscheweyh s integral operator [5 It is well known that Mocanu s class M of -convex functions is closed under Ruscheweyh s integral operator for >0

2 184 M A DURAI AND R PARVATHAM Theorem 1 Let fz= z + a 2 z 2 + be a uniformly -convex function in and let c>0 Then, for 0 < 1, the function Fz= F z,ζ F 0,ζ F 0,ζ, z,ζ, 4 is uniformly -convex where F z,ζ is defined as in 3 Proof We have from 3 that 1/ z,ζ = c +1/ z t ζ c 1 ft fζ 1/ dt 5 z ζ c ζ Differentiating with respect to z, we have z ζ c 1 F 1/ 1 z,ζf z,ζ+cz ζ c 1 1/ z, ζ = c + 1 z ζ c 1 fz fζ 1/ 6 and again differentiating with respect to z we get 1 1 z ζ 1/ 1 z,ζf z,ζ+z ζ 1 1/ 2 z,ζ F z,ζ 2 + 1/ 1 z,ζf z,ζ + c F 1/ 1 z,ζf z,ζ = c fz fζ 1/ 1 ; 1/ 1 z,ζf z,ζ z ζf z,ζ F z,ζ +1 z ζ F z,ζ F z,ζ +1+c 7 = c +1 fz fζ 1/ 1 Thus we get 1/ 1 z,ζf z,ζ z ζf [1+ z,ζ F z,ζ + z ζf z,ζ +c F z,ζ z ζf z,ζ F z,ζ 8 From 2 we have = c+1 fz fζ 1/ 1 showing that F0 = 0andF 0 = 1 Considering F z = F z,ζ F 0,ζ, 9 z ζf z Fz Fζ = z ζf z,ζ F z,ζ 10

3 A NOTE ON RUSCHEWEYH TYPE OF INTEGRAL OPERATORS 185 and differentiating with respect to z, we have F z F z + 1 z ζ F z Fz Fζ = 1 z ζ + F z,ζ F z,ζ F z,ζ F z,ζ ; 11 z ζf z +1 z ζf Z z ζf z,ζ = 1+ F z Fz Fζ F z,ζ F z,ζz ζ F z,ζ 12 Substituting 10 and 12 in8, we obtain 1/ 1 z,ζf z,ζ [ z ζf z F z +1 z ζf z + z ζf z Fz Fζ Fz Fζ +c = c+1 fz fζ 1/ 1 ; 1/ 1 z,ζf z,ζ 1 z ζf z Fz Fζ + 1+ z ζf z F z +c 13 = c+1 fz fζ 1/ 1 Setting Pz,ζ= 1 z ζf z Fz Fζ + z ζf z F z +1, 14 equation 13 becomes F 1/ 1 z,ζf z,ζ Pz,ζ+c = c+1 fz fζ 1/ 1 15 Taking logarithmic differentiation with respect to z, weget 1 z ζ F z,ζ F z,ζ +z ζf z,ζ F z,ζ + z ζp z,ζ Pz,ζ+c + = + z ζf z +1 z ζf z fz fζ ; [z ζ F z,ζ F z,ζ +1 z ζf z,ζ F z,ζ = 1+ z ζf z +1 z ζf z fz fζ + z ζf z,ζ + z ζp z,ζ F z,ζ Pz,ζ+c 16 Equations 10 and 12 give [ z ζf z +1 z ζf z F z Fz Fζ = 1+ z ζf z + z ζf z Fz Fζ + z ζp z,ζ Pz,ζ+c +1 z ζf z fz fζ 17

4 186 M A DURAI AND R PARVATHAM That is Hence, we have [ 1+ z ζf z +1 z ζf z F z Fz Fζ = 1+ z ζf z Pz,ζ+ z ζp z,ζ Pz,ζ+c = Since fz is uniformly -convex, we have Re +1 z ζf z fz fζ 1+ z ζf z Pz,ζ+ z ζp z,ζ Pz,ζ+c + z ζp z,ζ Pz,ζ+c z ζf z fz fζ 19 > 0 20 for all z,ζ, 0 1 We show that RePz,ζ>0 Suppose that there exists a point ζ 0 such that the image of the arc Γ : zt = ζ 0 + re it is tangent to the imaginary axis Let w 0 be the point of contact and let z 0 such that w 0 = Pz 0,ζ 0 ThenRePz 0,ζ 0 = 0 and therefore Pz 0,ζ 0 = ix, where x R Hence the outer normal to FΓ is z0 ζ 0 P z 0,ζ 0 = y<0 21 For such ζ 0, we have Re P z 0 ζ 0 P z 0,ζ 0 z 0,ζ 0 + P z 0,ζ 0 +c = Re = Re ix + y c+ix ix + yc ix c 2 2 +x 2 = c2 y < 0 for c>0 c 2 +x2 which contradicts 20 and hence RePz,ζ>0 in showing that Fz is a uniformly -convex function Acknowledgement This work was carried out when the first author was under the Faculty Improvement programme of University Grant and Commission of IX plan References [1 A W Goodman, On uniformly convex functions, Ann Polon Math , no 1, [2, On uniformly starlike functions, J Math Anal Appl , no 2, [3 S Kanas, Uniformly alpha convex functions, Int J Appl Math , no 3, [4 PTMocanu,Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica Cluj , French [5 S Ruscheweyh, Eine Invarianzeigenschaft der Basileviv Funktionen, Math Z , German M Anbu Durai: The Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai , India address: anbuduraim@yahoocoin R Parvatham: The Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai , India 22

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