Convolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations

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1 International Mathematical Forum, Vol., 7, no. 4, 59-7 HIKARI Ltd, Convolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations Khalida Inayat Noor Department of Mathematics COMSATS Institute of Information Technology Islamabad, Pakistan Bushra Malik Department of Mathematics COMSATS Institute of Information Technology Islamabad, Pakistan Syed Zakar Hussain Bukhari Department of Mathematics Mirpur University of Science and Technology (MUST) Mirpur-5(AJK), Pakistan Copyright c 6 Khalida Inayat Noor, Bushra Malik and Syed Zakar Hussain Bukhari. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A function analytic and locally univalent in a simply connected domain is of bounded radius rotation if its range has bounded radius rotation which is defined as the total variation of the direction angle of the radial vector to the boundary curve under a complete circuit. In this paper, we introduce some subclasses of analytic functions with bounded radius rotation involving subordination and establish integral and convolution preserving properties. We also determine estimates for the growth and distortion bounds, inclusions, conditions for starlikeness and bounds on coefficients differences. Most of our findings are related with the existing known results and several of their applications are found in literature of the subject.

2 6 K. I. Noor, B. Malik and S. Z. H. Bukhari Mathematics Subject Classification: Primary: 3C45, Secondary: 3C8 Keywords: Bounded radius rotation; Convolution; Schwarz function; subordination Introduction Let H(U) represent the class of all analytic functions f defined in the open unit disk U := {z C : z < }, and for a positive integer n and a C, let H[a, n] := { f H(U) : f(z) = a + a n z n + a n+ z n+ +..., z U }. Also, we denote A := {f H[, ] : f () = } The class of univalent functions is represented by S and it is a subclass of the class A, whereas, S, C, K and Q are the well-known classes of starlike, convex, close-to-convex and quasi-convex functions respectively. Let P denote the well-known class of Carathéodory functions p such that p H(U), with p() = and Re p(z) >, z U. Also P (β) represents the class of Carathéodory functions p such that p H(U) with p() = and Re p(z) > β, β <, z U. For details, we refer [4]. For f, g H(U), we say that f is subordinate to g and write as f(z) g(z), if there exists a Schwarz function w, that is, w H(U), with w() = and w(z) <, such that f(z) = g(w(z)) (z U). Using subordination defined above, Janowski [6] introduced the class P[A, B] for B < A. A function p analytic in U such that p() = belongs to the class P[A, B], if p(z) + Az + Bz (z U) or p(z) = + Aw(z) + Bw(z) (z U), where w is a Schwarz function. Geometrically, the image p(u) of p P[A, B] lies inside the open unit disk centered on the real axis with diameter ends at p( ) and p(). Clearly, P[A, B] P ( ) A B. Using a general bilinear fractional transformation T (z) = +A z, A +B z C, B [, ] and A B, Noor [8] introduced the class P[A, B ] consisting of functions p with p() = and p(z) +A z +B. Further extending this idea, Noor [9] introduced the class z P k [A, B] which is defined in the following.

3 Convolution and subordination properties 6 A function p analytic in U with p() =, belongs to the class P k [A, B], if and only if ( k p(z) = 4 + ) ( k p (z) 4 ) p (z), p, p P[A, B]. Also P k [, ] = P k and P k [ β, ] = P k (β). Pinchuk [8] studied the class P k, whereas Padmanabhan and Parvatham [6] introduced and investigated the class P k (β). For B < A and b C/{}, p P k [b, A,B] with p() =, if and only if there exists p P k [A, B] such that p(z) = bp (z) + ( b), z U. A function f belongs to the class V k [A, B], k, B < A, if and only if (zf (z)) f (z) P k [A, B] (z U). Similarly, f R k [A, B], if and only if zf (z) f(z) P k[a, B] (z U). For detail, we refer [, ]. We note that V k [ β, ] = V k (β) and R k [ β, ] = R k (β), β <. Definition.. Let f A. Then f T k,m [b, A, B; C, D], k, m, B < A, if and only if there exists g V m [C, D], C < D, such that { + ( )} f (z) b g (z) P k [A, B], b C/{} (z U). Definition.. Let f A. Then f Tk,m [b, A, B; C, D] for k, m, B < A and b C/{}, if and only if there exists g V m [C, D], D < C such that (zf (z)) P g k [b, A, B] (z U). (z) It is clear that f T k,m [b, A, B; C, D], if and only if zf T k,m [b, A, B; C, D]. For special choices, we refer Al-Amiri and Fernando [], Kaplan [7], Noor [3,, 3, 5], Polatoğlu and others [9 ] and Silvia [6] with references therein. Definition.3. Let f A. Then f M α k,m [b, A, B; C, D] for k, m

4 6 K. I. Noor, B. Malik and S. Z. H. Bukhari, α, b C/{} and B < A, if and only if there exists g V m [C, D], D < C such that { ( α) f (z) g (z) + α (zf (z)) } P g k [b, A, B] (z U). (z) It follows from the above definitions that f M α k,m[b, A, B; C, D] if and only if ( α) f + αzf T k,m [b, A, B; C, D]. () For special cases, see [3,, 3, 5] and others. To avoid repetition of parameters, we assume these parameters within the above specific ranges throughout our discussion. However, any change will obviously be mentioned. Preliminary Notes Lemma.[4]. If f C, g S, then for any analytic function F in U, (f F g) (U) (f g) (U) cof (U) (z U), where co F (U) denotes the closed convex hull of F (U). Lemma.[7]. Let N and D be analytic in U, D maps U onto a manysheeted starlike region N() = = D(), N () = D () = and N (z) D (z) P[A, B] (z U). P[A, B] (z U). Lemma.3. Let p P(b), b C/{} and h P k (β), k, β <, where p() = h() =. Then for z = r <, Then, N(z) D(z) (i) π π ( h(re iθ) dθ + [k ( β) )r r, (ii) π π ( p(re iθ) dθ + [4 b )r r. For the proof of part (i), we refer [4] and the part (ii) is proved in []. Lemma.4 Let f V k [A, B], k, B < A. Then for z = r <, ( Br) ( A B B )( k 4 + ) (+Br) ( A B B )( k 4 ), B exp ( A kr), B = f (z) (+Br) ( A B B )( k 4 + ) ( Br) ( A B B )( k 4 ), B exp (. A kr), B =

5 Convolution and subordination properties 63 The result is sharp for the extremal function: ( ( + Bz) ( k 4 + ) f (z) = ( Bz) ( k 4 ) ) A B B, B and f (z) = exp (A k ) z, for B =. Lemma.5 [5]. If g S [C, D], D < C, then so does ϕ g for ϕ analytic and convex in U. Lemma.6 []. Let β, n Z + and f V k [A, B] and F be defined by (F (z)) β = β + n z n t n (f(t)) β dt (z U). () Then f V ( A B ) = C ( A B ) for z < r, where r is given by. r = 3 Results and Discussion 4 k( B) + k ( B) + 6B. (3) In the following theorem, we study some integral preserving properties under certain assumption on parameters. Theorem 3.. Let f M α k,m [b, A, B; C, D]. Then F T k,m[b, A, B; C, D] such that for α >, f(z) = α z α t α F (t)dt (z U). (4). Proof. From (4),we can write ( α)f(z) + αzf (z) = F (z) (z U) and the result follows immediately by using () The following theorems deal with the convolution preserving properties under certain assumption on parameters. Theorem 3.. Let f T k, [b, A, B; C, D] and Ψ be any convex univalent function in U. Then Ψ f T k, [b, A, B; C, D] (z U). Proof. For G R [C, D] S [C, D], consider zf (z) G(z) G(z) z(ψ f) (z) (Ψ G)(z) = Ψ(z) Ψ(z) G(z) (z U).

6 64 K. I. Noor, B. Malik and S. Z. H. Bukhari From Lemma.4, we note that Ψ G S [C, D] and using Lemma., we have z(ψ f) (z) (Ψ G)(z) P k[b, A, B] (z U), where Ψ G S [C, D], which proves the required result. Theorem 3.3. Let f M α k, [b, A, B; C, D], α and Ψ be any convex univalent function in U. Then Proof. Using (), we have f Ψ M α k,[b, A, B; C, D] (z U). ( α)f(z) + αzf (z) T k, [b, A, B; C, D] (z U). In view of Theorem 3., we write which implies that {( α)f(z) + αzf (z)} Ψ(z) T k, [b, A, B; C, D], ( α)(f Ψ)(z) + αz(f Ψ)(z)) T k, [b, A, B; C, D] (z U). Again using relation (), we have the required result. Applications of Theorem 3.3 Let f M α k,m [b, A, B; C, D]. Then the functions Ψ i such that Ψ (z) = f(t) dt, t Ψ (z) = f(t)dt, Ψ 3 (z) = z f(t) f(xt) dt, x, x t xt and Ψ 4 (z) = + c z c t c f(t)dt, Re c > (z U) are also belong to M α k,m [b, A, B; C, D]. The proof is immediatly follows, when we observe that Ψ i (z) = (f ϕ i )(z), i =,, 3, 4 (z U), where ϕ i is convex in U such that ϕ (z) = log( z) = n= n zn, ϕ (z) = [z + log( z)] z ϕ 3 (z) = xz log[ x z ] and ϕ 4(z) = n= + c n + c zn, Re c > (z U). In the theorem below, we study inclusion with respect to the parameter α.

7 Convolution and subordination properties 65 Theorem 3.4. For α, we have M α k, [b, A, B; C, D] T k,[b, A, B; C, D]. Proof. For α =, the result is obvious. We assume that α > and using the integral representation for f M α k, [b, A, B; C, D], we have f(z) = α z α t α F (t)dt (z U), where F I k, [b, A, B; C, D] and f(z) = ϕ α (z) F (z). The function ϕ α such that ϕ α (z) = α(n ) + zn (z U) n= is convex. Using Theorem 3., we conclude that f T k, [b, A, B; C, D] and hence, we obtain the required result. In the next theorems, we determine the growth and distortion bounds and various other related results. Theorem 3.5. Let f T k,m [b, A, B; C, D]. Then for z = r < and D, we have and [ c Br ( Dr) ( m 4 + ) ( + Dr) ( m 4 ) ] C D D f (z) c + Br [ ( + Dr) ( m 4 + ) ( Dr) ( m 4 ) ( ) m R ( m ) exp Cr Br f (z) exp Cr R + Br, D =. ] C D D where c = k b ( Ar)+ b ( Br) and c = k b (+Ar)+ b (+Br). Proof. By Definition., there exists g V m [C, D] and p P k [b, A, B], such that f (z) = g (z)p(z) = g (z) [bh(z) + ( b)], h P k [A, B] [ {( k = g (z) b 4 + ) ( k p (z) 4 ) } ] p (z) + ( b), (5) where p, p P[A, B]. [ {( k f (z) g (z) b 4 + ) ( k p (z) + 4 ) } ] p (z) + ( b). Using distortion bounds for p, p P[A, B] proved in [6] and Lemma., we have the required result.

8 66 K. I. Noor, B. Malik and S. Z. H. Bukhari For k =, m = and b =, we obtain the bounds for function f such that f K[A, B; C, D], where the class K[A, B; C, D] is introduced by Silvia [6]. Theorem 3.6. Let f T,m [b, A, B; C, D] and F be defined by F (z) = n + z n t n f(t)dt, n Z + (z U). (6) Then F T, [b, A, B; β, ], for z < r, r is given by (3). Proof. Suppose that + ( ) f (z) b g (z) = p(z) (z U), where p P[A, B] and g V m [C, D]. Let us consider G(z) = n + z n t n g(t)dt, n Z + (z U). Then by using lemma.6 for α =, we have G C( C D ), for z < r, where r is given by (3). Consider + b ( ) F (z) G (z) = ( ) z b t n g (t)dt + b t n g (t)dt t n f (t)dt = N(z) D(z). Also N (z) D (z) = ( b )g (z) + f (z) b g (z) = ( + ) f (z) b f (z) P[A, B]. Now, since D is m-valently starlike, therefore by using Lemma., we have N(z) D(z) = + ( ) F (z) b G (z) P[A, B], for z < r. Hence, we have the desired proof. For b = and f T,m [, A, B; C, D] = T m [A, B; C, D], the function F defined by (6) is a close-to-convex function for z < r, where r is given by (3), for details, see []. Theorem 3.7. Let f T, [b, A, B; C, D]. Then the function F defined by () belongs to S (b).

9 Convolution and subordination properties 67 Proof. Let J(z) = t n (f(t)) β dt, β, n are positive integers. So from () on simplification, we have zf (z) F (z) = ( ) zj (z) nj(z) β J(z) Let + b ( ) zf (z) F (z) = ( )J(z) + (zj (z) nj(z)) b b βj(z) (z U). = N(z) D(z) : N() = = D(). By a result of Bernardi [] and the relation K[A, B; C, D] C proved by Silvia [6], we have D is (β + n )-valent starlike. Also [ N (z) D (z) = ( )J ( (z) + b b (zj (z)) nj (z) ) ] = + ( ) zf (z) β J (z) b f(z) P. Using Lemma. for A =, B =, we have N(z) D(z) = + ( ) zf (z) b F (z) P (z U). This proves the required result. Theorem 3.8. Let f T,m [b,, ; β, ] = T m (b; β). Then n a n+ a n e n ( k +)( β) ( 4 )( β) b (k( β) + ). Proof. We have F (z) = (zf (z)) = g (z) [H(z)p(z) + zp (z)], (H P k (β)). (7) As in [6], for g V k (β), there exists g V k such that g (z) = (g (z)) β ( ) ( k s (z) 4 + )( β) z ( s (z) z ) ( k 4 )( β), (s, s S ). Thus equation (7) becomes F (z) = (zf (z)) (s (z)) ( k 4 + )( β) = (z) ( β) (s (z)) [H(z)p(z) + ( k 4 )( β) zp (z)].

10 68 K. I. Noor, B. Malik and S. Z. H. Bukhari Since s is univalent, we choose z = z (r) with z = r, Q(r) = max z =r (z z )s (z) r, see [5] and write r π (z z )F (z) dθ Q(r) π ( s (z) ) [( k 4 + )( β)] ( H(z)p(z) + zp (z) ) dθ, π r π z ( β) s (z) ( k 4 )( β) where we have used the Schwarz inequality. Thus n a n (n+)z a n+ r π( r ) r (n+ β) π ( s (z) ) [( k 4 + )( β)] H(z)p(z) + zp (z) dθ. z ( β) s (z) ( k 4 )( β) Using the well-known distortion bounds for s, s S, see [4], we have n a n (n + ) z a n+ [ ] ( m 4 + )[( β) ] (4) ( m r n r ( r) 4 )( β) I, (8) where I = π π H(z) dθ π π p(z) dθ + r π π p (z) dθ. (9) Since p P(b), we can write p(z) = bh(z) + ( b), h P or p (z) = bh (z) and π π h (z) dθ, see [4]. () r Using Lemma.3 and inequality () in (9), we have [ ] + (m ( β ) )r [ ] + (4 b )r r b + r r r = [ ( + (m ( β ) )r ) ( + (4 b )r ) ] + r b. r This result along with (8) yields n a n (n + ) z a n+ [ ] ( m r 4 + )( β) r n+ ( + r) Ψ (m, r, b, β), ( r) where [ ( Ψ (m, r, b, β) = ( m )( β) + (m ( β ) )r ) ( + (4 b )r ) ] + r b. We take r = r n = ( n n+). Because z = r n, it follows that n a n a n+ e n ( m 4 + )[( β) ( 4) ( β) b [m( β) + ].

11 Convolution and subordination properties 69 4 Conclusion In this paper, we introduced some subclasses of analytic functions with bounded radius rotation involving subordination and established integral and convolution preserving properties. We also determined estimates for the growth and distortion bounds, inclusions, conditions for starlikeness and bounds on coefficients differences. Most of our findings are related with the existing known results found in literature of the subject. Acknowledgements. The authors would like to thank Honorable Rector CIIT, Islamabad Prof. Dr. SM Junaid Zaidi (HI, SI) and Worthy Vice Chancellor MUST, Mirpur Prof. Dr. Habib-ur-Rahman (FCSP, SI) for their untiring efforts for the promotion of research conducive environment in their relevant Institutions. References [] H.S. Al-Amiri and T.S. Fernando, On close-to-convex functions of complex order, Int. J. Math. Math. Sci., 3 (99), [] S.D. Bernardi, Convex and starlike univalent functions, Trans. Am. Math. Soc., 35 (969), [3] S.Z.H. Bukhari and K.I. Noor, Some subclasses of analytic functions with bounded Mocanu variation, Acta. Math. Hungarica, 6 (), [4] A.W. Goodman, Univalent Functions, Vol. I, II, Mariner Publ Comp. Tampa, Florida, 983. [5] G. Golusin, On distortion theorems and coefficients of univalent functions, Mat Sb., 9 (946), [6] W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math., 8 (973), [7] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., (95), [8] K.I. Noor, Applications of certain operators to the classes related with generalized Janowski functions, Integral Transforms Spec. Funct., (),

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13 Convolution and subordination properties 7 [] Y. Polatoğlu, M. Bolcal, A. Şen and E. Yavuz, A study on the generalization of Janowski functions in the unit disk, Acta. Math. Acad. Paedag. Nyíregyháziensis, (6), 7 3. [3] Ch. Pommerenke, On close-to-convex analytic functions, Trans. Am. Math. Soc., 4 (965), [4] S. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv., 48 (973), [5] H. Silverman and E.M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math., 37 (985), [6] E.M. Silvia, Subclasses of close-to-convex functions, Int. J. Math. Math. Sci., 6 (983), Received: December, 6; Published: January 4, 7