Cesáro partial sums of certain analytic functions

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1 JIA_30_edited [0/06 19:18] 1/8 Cesáro partial sums of certain analytic functions 1 abha W Ibrahim and Maslina Darus 1 Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur, Malaysia School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia 1 rabhaibrahim@yahoo.com maslina@um.my(corresponding author Abstract: The aim of the present paper is to consider geometric properties such as starlieness and convexity of the Cesáro partial sums of certain analytic functions in the open unit dis. By using the Cesáro partial sums, we improve some recent results including the radius of convexity. Keywords: analytic function; univalent functions; starlie functions; unit dis; Cesáro partial sums; partial sums AMS Mathematics Subject Classification: 30C45. 1 Introduction Let U := {z : z < 1} be a unit dis in the complex plane C and let H denote the space of all analytic functions on U. Here we suppose that H is a topological vector space endowed with the topology of uniform convergence over compact subsets of U. Also, for a C and n N, leth[a, n] bethe subspace of H consisting of functions of the form f(z =a + a n z n + a n+1 z n Further, let A := {f H: f(0 = f (0 1=0} and S denote the class of univalent functions in A. A function f Ais called starlie if f(u is a starlie domain with respect to the origin, and the class of univalent starlie functions is denoted by S. It is called convex C, if f(u is a convex domain. Each univalent starlie function f is characterized by the analytic condition ( zf (z > 0, z U. f(z Also, it is nown that zf (z is starlie if and only if f is convex, which is characterized by the analytic condition (1 + zf (z f > 0, z U. (z For a function f(z A, we introduce the partial sum of f(z by f (z =z + a n z n, z U. (1 n= For the partial sums f n (z off(z S, Szegö [1] showed that if f(z S,thenf n (z S for z < 1 4 and f n (z Cfor z < 1 8. Owa [] considered the starlieness and convexity of partial sums, f n (z =z + a n z n, 1

2 JIA_30_edited [0/06 19:18] /8 of certain functions in the unit dis. Moreover, Darus and Ibrahim [3] determined the conditions under which the partial sums of functions of bounded turning are also of bounded turning. In this paper, we consider the Cesáro partial sums, it is showed that this ind of partial sums preserve the properties of the analytic functions in the unit dis. obertson [4] showed that if f(z Ais univalent, then also all the Cesáro sums are univalent in the unit dis. Moreover, if its ordinary partial sums (1 is univalent in U, then the Cesáro sums are univalent. By employing the concept of the subordination, these results were extended by uscheweyha and Salinas [5]. The classical Cesáro means play an important role in geometric function theory (see [6-10]. From the partial sum s (z = a n z n, z U, with a 1 = 1, we construct the Cesáro means σ (z off Aby σ (z = 1 s n (z n=1 = 1 [ s1 (z+.. + s (z ] = 1 [ z +(z + a z +.. +(z a z ] = 1 [ z +( 1a z a z ] ( n +1 = z + an z n n= ( n +1 = f(z [z + z n ] n= = f(z g (z, n=1 where ( n +1 g = z + z n. n= Our aim is to consider geometric properties such as starlieness and convexity of the Cesáro partial sums of certain analytic functions in the open unit dis. Main results We define the function S which is a partial sum of f Aby S (z =z + ( a z,, a 0. ( Theorem 1 The function S (z satisfies for 1 a r 1 ( zs (z 1+ a r 1 (3 1 a r 1 S (z 1+ a r 1 0 r< 1 a 1, a 0.

3 JIA_30_edited [0/06 19:18] 3/8 Furthermore, S (z S (α for 1 α 0 r< 1 (1 α/ a 1, a 0. Proof. Noting that it follows that for cos θ 1, we obtain Moreover, we also observe that zs (z S (z ( zs (z = ( 1 S (z 1+ a r a r 1 = ( 1 1+ a z 1, 1+ a 1+ a cos θr 1 r 1 cos θ +( a r ( 1 Now assume that for This completes the proof. ( zs (z 1 a r 1 S (z 1. a r 1 1 a r 1 >α 1 a r 1 1 α 0 r< 1 (1 α/ a 1, a 0. emar For example, the values α =0.5, =and a = 1 imply the radius of starlieness of S (z isr = , and for the same values, the radius of starlieness of the ordinary partial sums f (z =z + a z is r = (see []. Next, we derive the radius of convexity. Theorem 3 The function S (z satisfies 1 a r 1 ( 1 a r 1 1+ zs (z S (z 1+ a r 1 1+ a r 1 (4 for Furthermore, S (z C(α for 1 0 r< 1 a 1, a 0. 1 α 0 r< 1 ( α a 1, a 0. Proof. A computation gives 1+ zs S (z ( 1 = (z 1+a z 1. 3

4 JIA_30_edited [0/06 19:18] 4/8 Therefore, for cos θ 1, we obtain (1+ zs (z S (z 1+ a cos θr 1 = ( 1 1+ a r 1 cos θ + a r ( 1 1+ a r 1 1+ a r 1. Moreover, we impose Now, consider that for This completes the proof. (1+ zs (z S (z 1 a r 1 1 a r 1. 1 a r 1 >α 1 a r 1 1 α 0 r< 1 ( α a 1, a 0. emar 4 In view of Theorem 3, for example, the values α =0.5, =and a =1posetheradius of convexity of S (z isr = and for the same values, the radius of convexity of the ordinary partial sums f (z =z + a z is r = (see []. Next, we assume special ordinary partial sums depending so that their coefficients satisfy the relation a n ( n+1. Theorem 5 Assume the partial sum Then the function f 3 (z S ( 1. Proof. We consider α such that ( zf 3 (z f 3 (z This implies that f 3 (z =z + 1 z + z3,. ( = 3 ( + 1 z z + z + 1 z z + z < 3 α, >α. that is, ( 1 1 z z + = r ( cos θ 1 z 1+ 1 r cos θ + r ( cos θ 1 < α. By letting t =cosθ, we define the function g(t as follows: g(t = r (t 1 rt + r (t 1. 4

5 JIA_30_edited [0/06 19:18] 5/8 Logarithmic derivative of g(t yields where g (t { (4t 1 g(t = { := = [1 1 [1 1 r [1 + 1 rt + [1 1 r (t 1][1 + 1 At + Bt + C r (t 1][1 + 1 r (t 1] + ( 1 r (t 1][1 + 1 h(t r +4 r t[1 1 r (t 1] rt + } rt + r (t 1] rt + r (t 1], r (t 1] } A =r 3 ( 1 B =4r [ 1 + ] C = 1 r[1 + 1 r ]. Now, for all andr 1, the function h(t has unique real negative zeros in the interval ( 1, 0. This leads to the fact that g (t has unique positive real zeros for all distributed in the interval (0, 1 Therefore, we will calculate α in t [ 1, 1. It is easy to chec that g(t is decreasing for r 1 in the interval [ 1, 1. Moreover, we have lim g(1 = lim = 3. We conclude that for t [ 1, 1, g(t <g( 1 3 = α, thus α = 1. This completes the proof. By letting = 3 in Theorem 5, we have the following result. Corollary 6 The Cesáro partial sums of the function f(z = σ 3 (z =z + 3 z z3, z U, z 1 z are starlie of order α = 1 5. Theorem 7 Assume the partial sum f 3 (z as in Theorem 5. Then the function f 3 (z C( 1 5. Proof. We consider α such that This implies that therefore, a computation gives (1+ zf 3 (z ( f 3 (z = 3 ( ( 1 z z +3 z z +1 ( z +3 z z +1 z +3 z < 3 α, = 1 ( 1 + ( ( 1 >α. z z +3( z,

6 JIA_30_edited [0/06 19:18] 6/8 thus 1+( 1 1 (1 3 r ( cos θ 1 r cos θ +3( ( cos θ 1 < α. By putting t =cosθ, we define the function G(t asfollows: G(t = Logarithmic derivative of G(t yields where 1+( 1 1 (1 3 r (t 1 rt +3r ( (t 1. G (t { G(t = [1r t][1 + ( 1 rt +3r ( (t 1] [1 3 r (t 1][1+( 1 rt +3r ( (t 1] [1r + t +r 1 ][1 3 r (t 1] } [1 3 r (t 1][1 + ( 1 rt +3r ( (t 1] { H(t } := [1 3 r (t 1][1+( 1 rt +3r ( (t 1] At + Bt + C = [1 3 r (t 1][1 + ( 1 rt +3r ( (t 1], 3 ( 1( A =1r B =1r [+3r ( ] C =r 1 [1+3r ( ]. Now, for all 3andr 1, the function H(t has unique real negative zeros in the interval [ 1, 0. This leads to the fact that G (t has unique positive real zeros for all 3 in the interval (0, 1 ]. So, we calculate α in the interval t ( 1, 1. A computation yields G(t is decreasing for r 1intheinterval t ( 1, 1. Thus, we have lim G(t < 9 10 = α, 1 >t>0.5, which implies α = 1 5. This completes the proof. Theorem 8 Assume the Cesáro partial sum of the function f(z = σ 3 (z =z z + z 3 Then the function σ 3 (z C( 1 for all 0. <r<0.5. Proof. We consider α such that z (1 z = z +z +3z (1+ zσ 3 (z σ 3 (z = (3 ( 4 3 z z >α. +3z 6

7 JIA_30_edited [0/06 19:18] 7/8 This implies that therefore, a computation gives thus ( 4 3 z z +3z ( 4 3 z z +3z < 3 α, = 1 ( 1 + (1 3z z, +3z 1 3r ( cos θ r cos θ +3(cos θ 1 < α. By putting t =cosθ, we define the function j(t asfollows: Logarithmic derivative of j(t yields j(t = 1 3r (t rt +3r (t 1. j (t { j(t := (t } [1 3r (t 1][ rt +3r (t 1] 16r 3 t +4r t = r(1+3r [1 3r (t 1][ rt +3r (t 1]. The function (t has a unique real negative zero in the interval t ( 1, 0 for all 0. <r<0.5 which is around t 1. This leads to the fact that j (t has a unique positive real zero in the interval (0, 1 around t 1. A computation yields j(t is decreasing in the interval t ( 1, 1 and assuming its maximums at t =0.5 andr =0.5. Thus, we have which implies α = 1. This completes the proof. lim j(t < 3 = α, r 0.5,t 0.5 Note that some other results related to partial sums can be found in [11], [1], [13], [14],[15]. Acnowledgement: The wor was fully supported by UKM-DLP and LGS/TD/011/UKM/ICT/03/0. The authors also would lie to than the referees for giving some suggestions for improving the wor. Competing interests The authors declare that they have no competing interests. Authors contributions Both authors jointly wored on deriving the results and approved the final manuscript. eferences [1] G. Szego, Zur theorie der schlichten abbilungen, Math. Ann. 100(198, [] S. Owa, Partial sums of certain analytic functions, IJMMS 5:1 ( [3] M. Darus,. W. Ibrahim, Partial sums of analytic functions of bounded turning with applications, Computational and Applied Mathematics, 9(1(010,

8 JIA_30_edited [0/06 19:18] 8/8 [4] M. S. obertson, On the univalency of Cesáro sums of univalent functions, National esearch Fellow, (1936, [5] S. uscheweyha, L. C. Salinas, Subordination by Cesáro means, Complex Variables and Elliptic Equations: An International Journal, 3:1 (1993, [6] Splina L.T., On certain applications of the Hadamard product, Appl. Math. and Comp., 199(008, [7] M. Darus,. W. Ibrahim, On Cesáro means for Fox-Wright functions, Journal of Mathematics and Statistics: 4(3: (008, [8] M. Darus,. W. Ibrahim, On some properties of differential operator, Acta Didactica Napocensia, : (009, pp 1-6. [9] M. Darus,. W. Ibrahim, Coefficient inequalities for concave Cesáro operator of nonconcave analytic functions, Eur. J. Pure Appl. Math, 3:6(010, [10] H.M. Srivastava, M. Darus,.W. Ibrahim, Classes of analytic functions with fractional powers defined by means of a certain linear operator, Integral Transforms and Special Functions 1:: (011, [11] B. A. Frasin, Generalization of partial sums of certain analytic and univalent functions, Appl. Math. Lett. 1 (008, [1] Z.-G. Wang, Z.-H. Liu and A. Catas, On neighborhoods and partial sums of certain meromorphic multivalent functions, Appl. Math. Lett. 4 (011, [13] G. Murugusundaramoorthy, K. Uma and M. Darus, Partial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta Function Abstract and Applied Analysis, Volume 011 (011, Article ID 84950, 9 pages. doi: /011/ [14] F. Ghanim and M. Darus, Partial sums of certain new subclasses for meromorphic functions. Far East J. Math. Sci., 55( (011, [15] abha W. Ibrahim and Maslina Darus, Partial sums for certain classes of meromorphic functions, Tamang J. Mathematics, Vol. 41 No. 1 (010,