Two Points-Distortion Theorems for Multivalued Starlike Functions

Transcription

1 Int. Journal of Math. Analysis, Vol. 2, 2008, no. 17, Two Points-Distortion Theorems for Multivalued Starlike Functions Yaşar Polato glu Department of Mathematics and Computer Science TC İstanbul Kültür University İstanbul, Turkey Emel Yavuz Department of Mathematics and Computer Science TC İstanbul Kültür University İstanbul, Turkey Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka, Osaka , Japan Yayoi Nakamura Department of Mathematics Kinki University Higashi-Osaka, Osaka , Japan Abstract Let A be the class of analytic functions fz) in the open unit disc U with f0) = 0 and f 0) = 1. Applying the fractional calculus for fz) A, the fractional operator D λ fz) is defined. Further, a new subclass S λ of A is considered using the fractional operator Dλ fz). The object of the present paper is to consider some properties of fz) in the class S λ.

2 800 Y. Polato glu, E. Yavuz, S. Owa and Y. Nakamura Mathematics Subject Classification: Primary 30C45 Keywords: subordination Analytic function, fractional calculus, fractional operator, 1 Introduction Let Ω denote the class of analytic functions wz) in the open unit disc U = {z C z < 1} with w0) = 0 and wz) < 1 for all z U. Further, let P be the class of analytic functions pz) of the form pz) =1+ p n z n z U) n=1 satisfying pz) = 1+wz) z U) 1 wz) for some wz) Ω. Let A be the class of functions fz) of the form fz) =z + a n z n that are analytic in U. For fz) in the class A, we introduce the definitions of fractional calculus fractional integrals and fractional derivatives) given by Owa [4], [5], or by Srivastava and Owa [6]. by Definition 1.1 The fractional integral of order λ is defined, for fz) A, D λ z fz) = 1 z Γλ) 0 fζ) dζ λ >0), z ζ) 1 λ where the multiplicity of z ζ) λ 1 is removed by requiring logz ζ) tobe real when z ζ) > 0. Definition 1.2 A, by The fractional derivative of order λ is defined, for fz) D λ z fz) = d dz Dλ z fz)) = 1 d Γ1 λ) dz z 0 fζ) dζ 0 λ<1), z ζ) λ

3 Two points-distortion theorems 801 where the multiplicity of z ζ) λ is removed by requiring logz ζ) tobe real when z ζ) > 0. Definition 1.3 Under the hypotheses of Definition 1.2, the fractional derivative of order n + λ) is defined, for fz) A,by and = Dz λ+n fz) = dn dz n Dλ zfz)) 0 λ<1,n N 0 = {0, 1, 2, }). It follows from the definitions for the fractional calculus that D λ z z k = D λ z z k = Γk +1) Γk +1+λ) zk+λ λ>0,k >0), Γk +1) Γk +1 λ) zk λ 0 λ<1,k >0) D n+λ z z k Γk +1) Γk +1 n λ) zk n λ 0 λ<1,k >0,n N 0,k n 1, 2, 3, ). Therefore, we say that, for any real λ, D λ z zk = Γk +1) Γk +1 λ) zk λ k>0,k λ 1, 2, 3, ). Moreover, using the fractional calculus, we define the fractional operator D λ fz) by D λ fz) = Γ2 λ)z λ D λ z fz) =z + If λ = 1, then D 1 fz) =Dfz) =zf z) and, if λ 2, 3, 4, and α 2, 3, 4,, then and D α D λ fz)) = D λ D α fz)) = z + Γn + 1)Γ2 λ) a n z n λ 2, 3, 4, ). Γn +1 λ) Γ2 λ)γ2 α)γn + 1)) 2 Γn +1 λ)γn +1 α) a nz n DD λ fz)) = zd λ fz)) = Γ2 λ)z λ λd λ zfz)+zd λ+1 z fz)).

4 802 Y. Polato glu, E. Yavuz, S. Owa and Y. Nakamura Now, let S λ be the subclass of A consisting of functions fz) which satisfy zd λ fz)) D λ fz) = pz) z U) for some pz) P. Let hz) Aand sz) A. Then hz) is said to be subordinate to sz), written by hz) sz), if there exists an analytic function wz) Ω such that hz) =swz)) z U). If sz) is univalent in U, then the subordination hz) sz) is equivalent to h0) = s0) and hu) su) see [1]). 2 Main Results In this section, we need the following lemma by Jack [2], also by Miller and Mocanu [3]. Lemma 2.1 Let wz) be a non-constant and analytic in U with w0) = 0. If wz) attains its maximum value on the circle z = r at a point z 1 U, then we have z 1 w z 1 )=kwz 1 ), where k is real and k 1. Lemma 2.2 If fz) Sλ, then r 1 + r) D λ fz) r 1) 2 1 r) 2 for z = r<1. Proof Using the definition for the operator D λ fz), we see that zd λ fz)) D λ fz) = pz) 1+z 1 z z U). It follows from the above that zd λ fz)) 1+r2 D λ fz) 1 r 2 or ) 1 r zd λ 1+r Re fz)) D λ fz) for z = r<1. 2r 1 r 2, 2) 1+r 1 r 3)

5 Two points-distortion theorems 803 On the other hand, since ) zd λ fz)) Re = r D λ fz) r log Dλ fz), using 3), we can write 1 r r1 + r) r log Dλ fz) 1+r r1 r). 4) Further, integrating the both sides of 4) from 0 to r, we can get 1). From the proof of Lemma 2.2, we have Corollary 2.3 The class S λ is compact. Proof After simple calculations from 2) we obtain z Dλ fz)) D λ fz) 1+r 1 r. 5) If we write the inequality 5) in the form D λ fz)) 1 r Dλ fz) 1+r 1 r, using Lemma 2.2 and after simple calculations we get D λ fz)) 1+r 1 r) 3. Then S λ is normal and therefore compact. Next, we derive Theorem 2.4 Let D λ fz) be an element of Sλ. If the function ) z + a azd λ f F λ z) =, a < 1, D λ fa)z + a)1 + āz) satisfies z F λ z) ) F λ z) 1 2z 1 z = Gz) 6)

6 804 Y. Polato glu, E. Yavuz, S. Owa and Y. Nakamura then F λ z) is starlike. This result is sharp because the extremal function is the solution of the fractional differential equation Γ2 λ)z λ D λ z fz) = z. 1 z) 2 Proof For ρ real, 0 <ρ<1, we define the function wz) by F λρ z) z )) z + a ad λ f ρ = D λ fρa)z + a)1 + āz) =1 wz)) 2, then wz) is analytic, w0) = 0 and z F λρ z) ) F λρ z) 1 = z1 a 2 ) z + a)1 + āz) [ + [ ρ z1 a 2 ) z + a)1 + āz) z z + a ) z + a D λ f ρ ))) z+a D λ f ρ z+a āz ]. )) 1 ] 7) Now, it is easy to realize that the subordination 6) is equivalent to wz) < 1 for all z U. Indeed, assume on the contrary: then, there exists a point z 1 U, max z = z1, such that wz) attains its maximum value on the circle z = r, at the point z 1, that is wz 1 ) = 1. Then by Lemma 2.1, we have z 1 w z 1 )=kwz 1 ) for some real k 1, which implies F λρ z z ) 1) 1 F λρ z 1 ) 1 = [ + z 1 1 a 2 ) z 1 + a)1 + āz 1 ) ρ z 1 1 a 2 ) z 1 + a)1 + āz 1 ) z 1 z 1 + a āz 1 1 = 2kwz 1) 1 wz 1 ) = Gwz 1)) / GU), ) ))) z1 + a D λ z f ρ 1 +a 1 )) 1 1 D λ z f ρ 1 +a 1 ] but this contradicts 6). Therefore our assumption is wrong, i.e, wz) < 1 for all z U, and we conclude that F λρ z) is starlike for every admissible ρ. From the compactness of S λ and 7) we infer that F λz) = lim ρ 1 F λρ z) is starlike. Corollary 2.5 If D λ fz) S λ, then u 1 λ 1 v 2 ) 2 v 1 λ 1 vu + u v ) 2 where z = u v 1 vu. D λ z fu) D λ zfv) u 1 λ 1 v 2 ) 2 v 1 λ 1 vu u v ), 2

7 Two points-distortion theorems 805 Proof If D λ fz) Sλ, then we have z 1 + z ) 2 Dλ fz) z 1 z ) 2. 8) Using the Theorem 2.4, the inequality 8) can be written the following form ) z + a z 1 + z ) azd λ f z 2 D λ fa)z + a)1 + āz) 1 z ). 9) 2 If we take a = v, u = z+a z = u v complete the proof of the corollary. Furthermore, we have 1 vu and after the brief calculations we Corollary 2.6 If D λ fz) Sλ, then 1 k 2 u u ) 2 1 k u ) kd λ z fu) 2 D λ z fku) 1 k 2 u 2 1 u ) k u ), 10) 2 1 k)u where z = 1 k u. 2 Proof If we take a = ku, u = z+a z = 1 k)u, then the inequality 9) 1 k u 2 can be written in the form 1 k 2 u u ) 2 1 k u ) 2 Using the definition of D λ fz) we obtain 10). Corollary 2.7 kd λ z fu) D λ zfku) 1 k 2 u 2 1 u ) k u ). 2 If D λ fz) Sλ, then 1 u 1 + u ) D λ z fu) 3 D λ z f u) 1+ u 1 u ) 3, where z = 2u 1+ u. 2 Proof Taking k 1 in 10), we obtain the result. Remark 2.8 The extremal function fz) in Theorem 2.4 is defined by Γ2 λ)z λ D λ z fz) = z 1 z) 2.

8 806 Y. Polato glu, E. Yavuz, S. Owa and Y. Nakamura Therefore, we have that ) fz) =D λ z D λ z fz)) = 1 z 1 λ Γ2 λ) D λ z 1 z) ) 2 1 = Γ2 λ) D λ z z 1 λ + nz n λ ) 1 Γ2 λ) nγn +1 λ) = z + z n Γ2 λ) Γ2) Γn +1) Γn +1 λ) = z + n 1)!Γ2 λ) zn = z + 2 λ) n 1 1) n 1 z n, where a) n denotes the Pochhammer symbol defined by { 1 n =0,a 0) a) n = aa + 1)a +2) a + n 1) n =1, 2, 3, ), so Γn +1 λ) =n λ)n λ 1) 2 λ) =2 λ) n 1. Γ2 λ) References [1] A. W. Goodman,Univalent Functions, Vol. I and Vol. II, Mariner Publishing Comp. Inc., Tampa, Florida, 1983 [2] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc ), [3] S. S. Miller and P. T. Mocanu,Differential Subordinations, Theory and Applications, Pure and Applied Math., Marcel Dekker, New York, [4] S. Owa, On the distortion theorems I., Kyungpook Math. J ), [5] S. Owa,Univalent and Geometric Function Theory Seminar Notes, TC İstanbul Kültür Univeristy Pub., İstanbul, [6] H. M. Srivastava and S. OwaEditors), Univalent Functions, Fractional Calculus and Their Applications, Jhon Wiley and Sons, New York, Received: February 19, 2008