Research Article Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions

Transcription

1 Complex Analysis Volume 13, Article ID 47596, 4 pages Research Article Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions Wasim Ul-Haq, Muhammad Arif, and Asfandyar Khan Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan Correspondence should be addressed to Wasim Ul-Haq; wasim474@hotmail.com Received May 13; Accepted 6 July 13 Academic Editor: Narendra Govil Copyright 13 Wasim Ul-Haq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In our present investigation, we introduce a subclass of analytic function associated with conic regions which is a form of generalized close-to-convexity. The arc-length inequality for a class of analytic function is well known. We derive this inequality for the newly defined class and also study some of its interesting consequences. 1. Introduction Let A denote the class of functions f: f (z) =z+ a n z n, (1) which are analytic in the unit disc U = z : z < 1}.Let S denote the class of all functions in A which are univalent. Also let S, C, andk be the well-known subclasses of A consisting of all functions which are, respectively, of starlike, convex, and close-to-convex. Kanas and Wisniowska [1, ] studied the classes of kuniformly convex denoted by k UCV and the corresponding class k ST related by the Alexandar type relation. Later Acu [3] considered the class k-uniformly close-to-convex denoted by k UK to be defined as k UK =f(z) A : Re ( zf (z) g (z) )>k zf (z) 1, g (z) n= g (z) k ST,z E}; () for more detail see [4 6]. In [7], the conic domain Ω k,γ with complex order is defined as Ω k,γ =γω k + (1 γ), < Re γ k+1, (3) Ω k =u+iv :u>k (u 1) + V }. (4) The domain Ω k,γ is elliptic for k>1, hyperbolic when < k<1,parabolicfork=1, and right half plane when k=. The functions which play the role of extremal functions for the conic regions of complex order are given as p k,γ (z) 1+(γ 1)z, 1 z k=, 1 1+ z (log π 1 z ), k=1, = 1 1 k sinh [( arccos k) arctan h z], π <k<1, 1 k 1 sin ( π u(z)/ t R (t) 1 dx) 1 x 1 (tx) k 1, k>1, (5) u(z) = (z t)/(1 tz), t (, 1), z E,andz is chosen such that k=cosh(πr (t)/4r(t)),r(t) is the

2 Complex Analysis Legendre s complete elliptic integral of the first kind and R (t) is complementary integral of R(t), see[1, ]. Let P=p(z):p()=1and Re p(z) >, z E} be the class of functions with positive real part, and let k P(γ) be the subclass of P containing the functions p(z),suchthat p(z) p k,γ (z). Motivated from Noor s work [8], we extend class k P(γ) to class k P m (γ), m which is defined as k P m (γ) = p (z) : p () =1and p 1 (z),p (z) in k P(γ) such that p (z) =( m )p 1 (z) ( m 4 1 } )p (z),z E }. (6) } Note that k P (γ) = k P(γ) and P m () = P m,theclass introduced and studied by Pinchuk [9]. We define the following class: k UT m (γ) =f(z) A : f (z) g (z) P,g(z) k UV m (γ), z E}, k UV m (γ) =f(z) A :1+ zf (z) f (z) k P m (γ), z E}. Geometrically, a function f(z) UV m (γ) means that the functional 1 + (zf (z)/f (z)) takes all the values in conic domain Ω k,γ and its boundary rotation is at most mπ.wenote that class k UT m (γ) coincides with already known classes of analytic functions by choosing special values for the involved parameters. For example, for k=,γ=1,wehavetheclass T m introduced and studied by Noor [1], and further along with this by taking m=,weobtainthewell-knownclassk of close-to-convex functions. The purpose of this paper is to investigate some interesting properties of class k UT m (γ). For this, we require the following results. Lemma 1. Afunctionf k UV m (γ) if and only if (7) (8) (i) f (z) = [f 1 (z)]γ/(1+k), f 1 (z) V m, (ii) there exist two normalized starlike functions s 1 (z) and s (z) such that f (z) = [ (s 1 (z) /z) ((k/4)+(1/)) γ/(1+k) (s (z) /z) ]. (9) ((k/4) (1/)) The above lemma can be proved by using the similar procedure as in [11]; also see [8]. Lemma (see [1]). Let h Pwith z=re iθ.then, 1 h (z) dθ 1+3r 1 r. (1). Some Properties of the Class k UT m (γ) In this section, we provide some of the interesting properties of class k UT m (γ) such as radius of convexity problem, arc length, and growth rate of its coefficients. The following theorem is readily seen when we proceed on similar lines as in [13]. Theorem 3. The function f(z) k UT m (γ) if and only if f (z) = (f 1 (z)) ((m/4)+(1/))γ, (11) ((m/4) (1/))γ (f (z)) f 1 (z) and f (z) are close-to-convex functions. Theorem 4. Let f k UT m (ρ, γ) in E. Then,f Cfor z < r, r = (1+k) mγ+ k+ + [mγ+k+] 4(1+k) (γ k 1). (1) This result is sharp. Proof. We can write f (z) =g (z) h (z), Using Lemma 1(ii), we get g(z) k UV m (γ), h (z) P. (13) f (z) =[ (s 1 (z) /z) ((m/4)+(1/)) γ/(1+k) (s (z) /z) ] h (z), (14) ((m/4) (1/)) s 1 and s are starlike functions. Logarithmic differentiation of (14)givesus zf (z) f (z) = γ 1+k [ 1 + (m ) zs 1 (z) s 1 (z) (m 4 1 ) zs (z) s (z) ] + zh (z) h (z). (15)

3 Complex Analysis 3 This implies that 1+ zf (z) f (z) = 1+k γ 1+k 1+k [(m ) zs 1 (z) s 1 (z) ( m 4 1 ) zs (z) s (z) ]+zh (z) h (z). Now using distortion results for the class P,we have Re (1 + zf (z) f (z) ) 1+k γ 1+k 1+k [(m ) 1 r 1+r ( m 4 1 ) 1+r 1 r ] r 1 r (16) = (1+k γ)(1 r )+γ[1+r mr] r(1+k). (1+k)(1 r ) (17) The right hand side of (17)ispositivefor z < r,r is given by (1). The sharpness can be viewed from the function f k UT m (γ),givenby f (1+z)(γ/(k+1))(((m/) 1)+1) (z) =, z E. (18) (γ/(k+1))(((m/)+1)+1) (1 z) We note the following interesting special cases: (i) For γ=1, we have the radius of convexity for class k UT m. (ii) For γ=1and k=, we have the radius of convexity for class T m,provedbynoor[1]. (iii) For γ = 1, k = and m =,wehaveradiusof convexity for close-to-convex functions which is well known. Theorem 5. Let f k UT m (γ) with k, m,and ((m + )/(1 + k)) Re γ>1.then, L r (f) A (k, γ, m) ( 1 (1/)((m+)/(1+k)) Re γ 1 r ). (19) The exponent (1/)((m + )/(1 + k)) Re γ is sharp. Proof. Let f k UT m (ρ, γ). Then, there exists g(z) k UV m (γ) such that f (z) =g (z) h (z), h P. () From the definition of k UV m (γ), onecandeducethat g(z) k UV m (1) implies that g(z) V m (k/(k + 1)). Now using (), Lemma 1(ii), and distortion theorems for starlike functions, we have L r (f) = zg (z) h (z) dθ, g (z) k UV m (γ), h (z) P = z (s 1 (z) /z) ((m/4)+(1/))(1/(1+k))γ (s (z) /z) ((m/4) (1/))(1/(1+k))γ h (z) dθ = z 1 γ(1/(k+1)) (s 1 (z)) ((m/4)+(1/))(1/(1+k))γ (s (z)) ((m/4) (1/))(1/(1+k))γ h (z) dθ ((m/) 1)(1/(k+1)) Re γ r ((m/4)+(1/))(1/(k+1)) Re γ 1 s 1 (z) (1/4)((m+)/(1+k)) Re γ h (z) dθ. By using Hölder s inequality, this gives L r (f) ((m/) 1)(1/(k+1)) Re γ r ((m/4)+(1/))(1/(k+1)) Re γ 1 ( 1 ( 1 s 1 (z) (1/)((m+)/(1+k)) Re γ 1/ dθ) 1/ h (z) dθ). (1) () Since ((m + )/(1 + k)) Re γ>1, therefore subordination for starlike functions and Lemma give us L r (f) A (k, γ, m) ( 1 (1/)((m+)/(1+k)) Re γ 1 r ). (3) The function F (z) k UT m (γ) is defined by G (1+z)((m/) 1)(1/(1+k))γ (z) = (1 z), ((m/)+1)(1/(1+k))γ shows that the exponent is sharp. F (z) =G (z) h (z), (4) h (z) = 1+z 1 z (5) Some special choices in the above theorem give us the following interesting results. Corollary 6. Let f k UT (1).Then L r (f) A (k) ( 1 /(1+k) 1 r ). (6) Corollary 7. Let f(z) T m.then L r (f) A (m) ( 1 1 r ) (m/)+1. (7)

4 4 Complex Analysis Coefficient Growth Problems. The problem of growth rate and asymptotic behavior of coefficients is well known. In the next results, we study these problems for class k UT m (γ) by varying different parameters. Theorem 8. Let f k UT m (γ) with k, m and ((m + )/(1 + k)) Re γ>1.then a n =O(1) n((m/)+1)(re γ/(1+k)) 1, (n ). (8) The exponent is sharp. Proof. With z=re iθ,cauchy stheoremgivesus na n = 1 r n zf (z) dθ = 1 r n L r (f), z = re iθ. (9) Using Theorem 5 and putting r = 1 (1/n), weobtainthe required result. The sharpness follows from the function F defined by the relation (4). [7] K. I. Noor, M. Arif, and W. Ul-Haq, On k-uniformly close-toconvex functions of complex order, Applied Mathematics and Computation,vol.15,no.,pp ,9. [8] K. I. Noor, On a generalization of uniformly convex and related functions, Computers & Mathematics with Applications,vol.61, no. 1, pp , 11. [9] B. Pinchuk, Functions of bounded boundary rotation, Israel Mathematics,vol.1,pp.6 16,1971. [1] K. I. Noor, On a generalization of close-to-convexity, International Mathematics and Mathematical Sciences,vol.6, no., pp , [11] M. K. Aouf, On certain classes of p-valent functions, International Mathematics and Mathematical Sciences,vol.9, no. 1, pp , [1] Ch. Pommerenke, On close-to-convex analytic functions, Transactions of the American Mathematical Society,vol.114,pp , [13] K. I. Noor, On certain analytic functions related with strongly close-to-convex functions, Applied Mathematics and Computation,vol.197,no.1,pp ,8. Corollary 9. Let f k UT m (1), andletitbeoftheform (1).Then,forn>3, k,onehas a n =O(1) n((m/)+1)(1/(1+k)) 1. (3) For k =, in the above corollary, we have growth rate of coefficients problem for functions in class T m,and,fork=, m=gives us the growth rate of coefficients for close-to-convex functions, which are well known. Acknowledgments The authors want to acknowledge worthy referees of this paperfortheirinsightfulcommentswhichgreatlyimproves the entire presentation of the paper. They would also like to thank Prof. Dr. Ehsan Ali, VC AWKUM, for providing research facilities. References [1] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, Computational and Applied Mathematics, vol. 15, no. 1-, pp , [] S. Kanas and A. Wisniowska, Conic domains and starlike functions, Revue Roumaine de Mathématiques Pures et Appliquées, vol. 45, no. 4, pp ,. [3] M. Acu, On a subclass of n-uniformly close to convex functions, General Mathematics,vol.14,no.1,pp.55 64,6. [4] A. Gangadharan, T. N. Shanmugam, and H. M. Srivastava, Generalized hypergeometric functions associated with kuniformly convex functions, Computers & Mathematics with Applications, vol. 44, no. 1, pp ,. [5] E.Aqlan,J.M.Jahangiri,andS.R.Kulkarni, Newclassesofkuniformly convex and starlike functions, Tamkang Mathematics,vol.35,no.3,pp.61 66,4. [6] S.KanasandH.M.Srivastava, Linearoperatorsassociatedwith k-uniformly convex functions, Integral Transforms and Special Functions,vol.9,no.,pp.11 13,.

5 Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization