Some Classes of k-uniformly Functions with Bounded Radius Rotation

Transcription

1 Appl. Math. Inf. Sci. 8 No Applied Mathematics & Information Sciences An International Journal Some Classes of k-uniformly Functions with Bounded Radius Rotation Khalida Inayat Noor Rabia Fayya and Muhammad Aslam Noor Mathematics Department COMSATS Institute of Information Technology Park Road Islamabad Pakistan Received: 4 Mar. 3 Revised: 4 Jul. 3 Accepted: 6 Jul. 3 Published online: Mar. 4 Abstract: We use Ruscheweyh derivative to define certain new classes of analytic functions with bounded radius rotation and related to conic domains. Some interesting and significant results such as inclusion results growth rate of coefficients and radius problems for these new classes of k-uniformly functions. Several special cases are discussed. Results obtained in this paper may stimulate further research activities in this field. Keywords: Convex starlike conic domains subordinate Bernardi operator bounded radius rotation convolution radius of convexity. AMS Subject Classification: 3C45 3C5 3C55 Introduction Let A be the class of functions of the form f=+ a j j which are analytic in the open unit disc E = : <}. Let S δ and Cδ denote the subclasses of A consisting of starlike and convex functions of order δ δ < respectively where S = S and C = C are the well known classes of starlike and convex functions. Let S A be the class of univalent functions. Then the inclusion relation C S S holds e.g. see [3]. Kanas and Wisniowska [5] studied the classes of k-uniformly convex functions denoted by k UCV and the corresponding class k ST related by the Alxander type relation. See also []. Let fg A g = + b j j and f given by.. Then the convolution Hadamard product of f and g is defined by f g=+ a j b j j =g f E. Also if f and g are analytic in E we say that f is subordinate to g in E written as f g or f g if there exists a Schwar function w such that f=gw for E. For k [ the conic domain Ω k are defined as follows see [4]. Ω k =u+iv : u>k u + v }. For fixed kω k represents the conic regions bounded successively by the imaginary axis k= the right branch of a hyperbola <k< and a parabola k=. When k > the domain becomes a bounded domain being interior of the ellipse. We shall consider here k []. Related to the domains Ω k the following functions p k play the role of extermal functions mapping E onto Ω k. These functions are univalent in E and belong to the class P of functions with positive real part and are given as: p k = + k= + π log + k= + k sinh[ π arccosk arctan ] <k<. 3 Using subordination concept we define the class Pp k as follows. Let p be analytic in E with p =. Then p Pp k if p p k k [] and p k are given by.3. Corresponding author khalidanoor@hotmail.com

2 58 K. I. Noor et al: Some Classes of k-uniformly Functions... It is known [5] that p Pp k is in the class Pδ of functions with positive real part greater than δ = k+ k. That is Pp k P k k+ P. We generalie the class Pp k as follows. Definition. Let p be analytic in E with p =. Then p P m p kα m α < k [] if and only if we can write m p = αp +α} where p i Pp k i=. αp +α} 4 Also it is obvious that p P m p kα can be written as p= αh+α h P m p k =P m p k in E. For m = α = we have the class Pp k. When k = the class P m p α reduces to P m α see [] and P m =P m was introduced in []. The relation 4 can be expressed as: p = 4 + p φ α 4 p φ α 5 where φ α = α Rφ α α E and p i p k i=. For n N = 3...} let D n : A A be the operator defined by D n f = f n+ so that D n f = n f n n! j+ n! = + n! j! a j j. The following identity holds and can easily be verified. D n f =n+d n+ f nd n f. 6 The operator D n is called Ruscheweyh derivative of order n see [6]. We shall assume unless otherwise stated that n N m k [] α < and E. A function f A is said to belong to the class R m nα if and only if D n f D n f P m α E. When n = α = we get the class R m of functions with bounded radius rotation see [3]. We now define the following. Definition. For n N m α < k [] a function f A is said to belong to the class k UR m nα if and only if for E. D n f D n f P m p k α As special cases we have the following: i. UR m kα=r m kα. ii. UR m =R m. iii. k UR =k ST. By using Alexander type relation the class k UV m nα is defined as follows. Let f A. Then if and only if. We note that f k UV m nα f k UR m nα E i. UV m nα= V m nα. ii. For k = n = α = we have the class k UV =k UCV. V m of the functions with bounded boundary rotation see [3]. For a function f A we define the integral operator I c : A A by I c f= c+ c t c ftdt Rc>. 7 When c N = 3...} the operator I c was introduced by Bernadi []. In particular I was studied earlier by Libera [6] and Livingston [7]. Preliminaries In order to derive our main results we need the following lemmas. Lemma [4] p 7. Let β > γ and let h be analytic in E with h=. Then h φ βγ = h+ h βh+γ 8 where φ βγ = j= β + γ j. 9 jβ + γ

3 Appl. Math. Inf. Sci. 8 No / 59 Lemma. Let k and let σδ be any complex numbers with σ and α < R σk k+ + δ. If h is analytic in E h= and satisfies h+ h p kα σh+δ and q kα is an analytic solution of q kα + q kα σq kα +δ = P kα then q kα is univalent and h q kα p kα. The function q kα is the best dominant of and is given as t q kα = p kα u exp du dt δ u σ. t This result can be found in [] and is an easy generaliation of one due to Kanas [4]. Lemma 3[3]. Let f C and g S. Then for every function F analytic in E with F= we have f Fg E CoFE f g where CoFE denotes the closed convex hull of FE. Lemma 4[5]. Let p be an analytic function in E with p= and Rp> E. Then for s> and ν Complex } R p+ sp > p+ν for <r where r is given by r = ν+ A+A ν A=s+ + ν and this radius is best possible. 3 Main Results Theorem. k UR m n+α k UR m nα for each n N. Proof. Let f k UR m n+α. Then for E D n+ f D n+ f P m p kα. Set D n f D n f = H. H is analytic in E and H=. From 6 and we obtain D n+ f } D n+ = H+ H P m p kα in E. f H+n Let H= 4 + h 4 h. 3 Using 3 and Lemma with α = c=n it follows that h i + h i Pp kn i= E. 4 h i +n Applying Lemma we obtain h i Pp kα in E for i = and consequently H P m p kα in E. This proves that f k UR m nα in E. Corollary. Let k=. Then f R m n+α implies that f R m nσ where σ is given by [ ] n+ σ = F αn+; n 5 and F represents Gauss hypergeometric function. This result is sharp. Proof. In fact from 4 it follows that h i + h i Pα i=. h i +n This implies by using a result due to Miller-Mocann [8 p 3 Theorem 3.3 e] that h i Pσ where σ is given by 5. For sharpness the extremal function is given as with p = g n α g = t n dt t = F αn+. n+ Consequently H P m σσ is given by 5. This completes the proof that f R m nσ in E.

4 53 K. I. Noor et al: Some Classes of k-uniformly Functions... As a special case when n = we note that R m α = V m α where V m α is the class of functions of bounded boundary rotation with order α see [3] and f V m α implies that f R m σ=r m σ where R m σ is the corresponding class of bounded radius rotation with order σ order σ is given by 5 with n= as follows. σ = σ =[ F α; ] α α α α = = ln. α = When α = we have σ = =. This shows if f V m then f R m. When m= this leads to a well known result that a convex univalent function is a starlike function of order. Since Pp kα P k+α k+ we have the following result on the rate of growth of coefficients for f R m nγ γ = k+α k+. Theorem. Let f R m nγ γ = k+ k+α and be given by. Then for j>3 m. a j = O. j γ + n+} where O is a constant depending on mkα and n. The exponent γ + n+} is best possible. Proof. D n f = f n+ [ ] [ j+ n! = + n! j! j + = + j+ n! n! j! a j j. a j j ] Now since D n f V m γ if and only ifd n f R m γ we use a coefficient result for the class V m γ proved in [9] to have for j> 3m. j+ n! n! j! a j <m γ + m γ} γ 3 j γ + and this gives us a n = O. j γ + n+} j> 3m. The function F R m nγ defined by D n F = +δ γ δ + γ δ = δ = shows that the exponent γ + n+} is the best possible. Theorem 3. n= R m nγ=id} γ = k+α k+ where id is the identity function. Proof. Let f =. Then it follows trivially that R m nγ for n N. On the contrary assume that f R m nγ with f given by. Then from n= Theorem we deduce that f=. We now show that the class k UR m nα is preserved under generalied Bernardi operator given by 7. Theorem 4. Let f k UR m nα and let I c frc > be defined by 7. Then for E I c f k UR m nα. Proof. Let D n I c f D n I c f = h= 4 + where h is analytic in E h=. p 4 p 6 Simple computations and use of 7 6 lead to the following. D n f D n = h+ h P m p kα. 7 f h+c With similar technique used in Theorem and from 6 7 it follows that p i + p i Pp kα Ei=. p i +c We now apply Lemma to have p i Pp kα i= and consequently h P m p kα E. This proves I c f k UR m nα. As special cases we note that: i. For m = n = = αthe class k ST is preserved under generalied Bernardi operator. ii. With m= n= α = it follows from Theorem 4 that the class k UCV of uniformly convex functions is invariant under the operator given by 7. Corollary. Let f k UR m nα R m nα α = k+ k+α. Then I c f R m nβ where c> and } c+ β = F α ;c+; c. 8

5 Appl. Math. Inf. Sci. 8 No / 53 Proof. Proceeding as in Theorem 4 it follows from 7 that p i + p i p i +c Pα α = k+ α k+. Then as in corollary p i Pβi = E where β is given by 8. Consequently from 8 h P m β and hence I c f R m nβ E. Corollary 3. i. Let α = k= and c=. Then f UR m n R m n. Then from Theorem 4 I f R m nβ where β = = F ln. ii. For c= k==α I f R m nβ where } β = F ;3 = ln. iii. For c = k = α = f UR m n R m n. This implies I f R m nβ where } β = F ;3 = ln.69. Theorem 5. Let f k UR m nα. Then I n f k UR m n+α where I n f is defined by 7 with c = n N. The proof is straightforward when we note from 6 and 7 that D n f=d n+ I n f. Next we consider the converse of the problem involving the operator 7 for the case when m=. Theorem 6. Let I c f defined by 7 belong to k UR nα. Then f k UR nα for <r c where 3+c r c = c c 9 c= and this radius is best possible. Proof. We first prove the following. Let f k UR nα and let φ be convex univalent in E. Then we show that φ f k UR nα in E. Now D n φ f D n φ f = φ Dn f φ D n f = φ Dn f D n f.d n f φ D n f = φ FDn f φ D n f. Since f k UR nα D n f S k+α k+ S F Pp kα and φ C in E we use Lemma 3 to conclude that φ f k UR nα. We define h c by h c = j= j+ c +c j = [ c ] +c. Now from 7 we can write Then f =I c f= c+ c t c ftdt f= ci c f+i c f c+ c>. = I c f h c where h c is given by. It is known [] that h c is convex for <r c where r c is given by 9. Since I c f k UR nα in E and h c is convex for < r c it follows that for f k UR nα for <r c. As special cases we note that: i. For c = I f is Libere-Livingston operator and I f k UR nα in E implies that f k UR nα for <. ii. Alexander operator I f k UR nα in E implies that f k UR nα for < 3. Remark. Let f n = n N n+. Then it can easily be verified that f n C for <r n where r n = 3n++ 3n+ 4n. We have: Theorem 7. Let D n f k UR α = k STα in E. Then f k STα for <r n where r n is given by. Proof. Since D n f k UR α=k STα it follows that f k STα. n+ Now is convex for < r n+ n and k STα is closed under convex convolution we obtain the required result. Theorem 8. Let I n f be defined by 7 with c=n N and let I n f k UR nα. Then for γ = k+ k+α Dn f S γ for <r γ where the exact value of r γ will be given in the proof. Proof. Proceeding as in Theorem 4 with n=c we have D n f D n f = h+ h h+n

6 53 K. I. Noor et al: Some Classes of k-uniformly Functions... where h= Dn I n f D n I n f Pp kα Pγ in E. From we have with h= γh +γ. D n f } γ D n γ f = h + γ.h h + n+γ h P.3 γ We use Lemma 4 with s= γ ν = n+γ γ to have [ D n f }] R γ D n γ f or <r γ f where This completes the proof. r γ = ν+ A+A ν A=s+ + ν s= γ ν = n+γ γ. Remark. We can use well known distortion results for the class P see[3] to have [ D n f }] R γ D n γ f } Rh γ r γ r +r +n+r That is R [ D n f γ D n f Rh }] γ γ nr γr+n+ γ r +n+γ r and right hand side is positive for r<r where r = } n+ γ+ γ n+ γ n. 4 References [] R. W. Barnard C. Kellogg Applications of Convolution operators to problems in Univalent Function Theory Michigan Math. J [] S. Bernardi Convex and starlike univalent functions Trans. Amer. Math. Soc [3] A. W. Goodman Univalent Functions Polygonal Publishing House Washington New Jersey III 983. [4] S. Kanas Techniques of differential sunordination for domains bounded by conic sections Internat. J. Math. Math. Sci [5] S. Kanas A. Wisniowska Conic regions and k-uniform convexity J. Comput. Appl. Math [6] R. J. Libera Some classes of regular univalent functions Proc. Amer. Math. Soc [7] A. W. Livingston On the radius of univalence of certain analytic functions proc. Amer. Math. Soc [8] S. S. Miller P.T. Mocanu Differential Subordinations: Theory and Applications series on Monographs and Textbooks in Pure and Applied Mathematics Marcel Dekker Inc. New York Basel 5. [9] K. I. Noor On subclasses of close-to-convex functions of higher order Internat. J. Math. Math. Sci [] K. I. Noor M. Arif Wasim-ul-Haq On k-uniformly close-to-concex functions of complex order Appl. Math. Comput [] K. S. Padmanabhan R. Parvatham Properties of a class of functions with bounded boundary rotation Ann. Polon. Math [] B. Pinchuk Functions of bounded boundary rotation Israel J. Math [3] S. Ruscheweyh Convolution in Geometric Function Theory Les Presse. De universite de Montreal Montreal 98. [4] S. Ruscheweyh Ein Invarianeign- shaft der Bailevic- Funkionen Math. Z [5] S. Ruscheweyh V. Singh On certain extremal problems for functions with positive real parts Amer. Math. Soc [6] S. Ruscheweyh A new criteria for univalent functions Proc. Amer. Math. Soc This shows that for D n f S γ for <r and r is given by 4. Acknowledgement The authors are grateful to Dr. S. M. Junaid Zaidi Rector COMSATS Institute of Information Technology Pakistan for providing excellent research and academic environment.

7 Appl. Math. Inf. Sci. 8 No / Khalida Inayat Noor is a leading world-known figure in mathematics and is presently employed as HEC Foreign Professor at CIIT Islamabad. She obtained her PhD from Wales University UK. She has a vast experience of teaching and research at university levels in various countries including Iran Pakistan Saudi Arabia Canada and United Arab Emirates. She was awarded HEC best research paper award in 9 and CIIT Medal for innovation in 9. She has been awarded by the President of Pakistan: Presidents Award for pride of performance on August 4 for her outstanding contributions in mathematical sciences and other fields. Her field of interest and specialiation is Complex analysis Geometric function theory Functional and Convex analysis. She introduced a new technique now called as Noor Integral Operator which proved to be an innovation in the field of geometric function theory and has brought new dimensions in the realm of research in this area. She has been personally instrumental in establishing PhD/MS programs at CIIT. Dr. Khalida Inayat Noor has supervised successfully several Ph.D and MS/M.Phil students. She has been an invited speaker of number of conferences and has published more than 4 Four hundred research articles in reputed international journals of mathematical and engineering sciences. She is member of editorial boards of several international journals of mathematical and engineering sciences. Muhammad Aslam Noor earned his PhD degree from Brunel University London UK 975 in the field of Applied MathematicsNumerical Analysis and Optimiation. He has vast experience of teaching and research at university levels in various countries including Pakistan Iran Canada Saudi Arabia and United Arab Emirates. His field of interest and specialiation is versatile in nature. It covers many areas of Mathematical and Engineering sciences such as Variational Inequalities Operations Research and Numerical Analysis. He has been awarded by the President of Pakistan: President s Award for pride of performance on August 4 8 in recognition of his contributions in the field of Mathematical Sciences. He was awarded HEC Best Research paper award in 9. He has supervised successfully several Ph.D and MS/M.Phil students. He is currently member of the Editorial Board of several reputed international journals of Mathematics and Engineering sciences. He has more than 75 research papers to his credit which were published in leading world class journals. Rabia Fayya is lecturer in Department of Mathematics COMSATS Institute of Information and Technology Islamabad Pakistan. Her field of interests are Complex Analysis Geometric Function Theory and related areas.