Research Article On Generalized Bazilevic Functions Related with Conic Regions

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1 Applied Mathematics Volume 1, Article ID 9831, 1 pages doi:1.1155/1/9831 Research Article On Generalized Bazilevic Functions Related with Conic Regions Khalida Inayat Noor and Kamran Yousaf Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan Correspondence should be addressed to Khalida Inayat Noor, khalidanoor@hotmail.com Received 1 March 1; Accepted 18 March 1 Academic Editor: Yonghong Yao Copyright q 1 K. I. Noor and K. Yousaf. 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. We define and study some generalized classes of Bazilevic functions associated with convex domains. These convex domains are formed by conic regions which are included in the right half plane. Such results as inclusion relationships and integral-preserving properties are proved. Some interesting special cases of the main results are also pointed out. 1. Introduction Let A denote the class of analytic functions f defined in the unit disc E z : z < 1} and satisfying the conditions f, f 1. Let S denote the subclass of A consisting of univalent functions in E,andletS and C be the subclasses of S which contains, respectively, star-like and convex in Bazilevič 1 introduced the class B α, β, h, g as follows. Let f A. Then, f B α, β, h, g, α, β real and α>if [ (α ) z 1/ α iβ f iβ hg α t t dt] iβ 1, 1.1 for some g S and Re h >,z E. The powers appearing in 1.1 are meant as principle values. The functions f in the class B α, β, h, g are shown to be analytic and univalent, see 1. B α, β, h, g is the largest known subclass of univalent functions defined by an explicit formula and contains many of the heavily researched subclasses of S. We note the following: i B 1,, 1,g C, ii B 1,,zg /g, g S,

2 Applied Mathematics iii B 1,,h,g K, where K is the class of close-to-convex functions introduced by Kaplan, iv B cos γ,sin γ,cos zg / g i sin γ,g is the class of γ-spiral like functions which are univalent for γ <π/. For analytic functions f n a nz n and g n b nz n,byf g we denote the Hadamard product convolution of f and g, defined by ( ) f g a n b n z n. n 1. For k,, the conic domain Ω k is defined in 3 as follows: Ω k } u iv : u>k u 1 v. 1.3 For fixed k, Ω k represents the conic region bounded successively by the imaginary axis k, the right branch of hyperbola <k<1, a parabola k 1 and an ellipse k >1. The following univalent functions, defined by p k with p k 1andp >, map k the unit disc E onto Ω k : 1 z, k, 1 z 1 ( log 1 ) z π 1, k 1, z p k 1 1 k sin h[ A k arc tanh z ], <k<1, ( ( )) π z 1 k 1 sin K t F t,t, k >1, 1.4 where A k /π arc cos k, F w, t is the Jacobi elliptic integral of the first kind: F w, t w dx 1 x 1 t x, 1.5 and t, 1 is chosen such that k cosh πk t /K t, where K t is the complete elliptic integral of the first kind, K t F 1,t,K t K 1 t. It is known that p k are continuous as regards to k and have real coefficients for k,. Let P p k be the subclass of the class P of Caratheodory functions p, analyticine with p 1 and such that p is subordinate to p k, written as p p k in E. We define the following.

3 Applied Mathematics 3 Definition 1.1. Let h be analytic in E with h 1. Then, h P m p k if and only if, for m,k,,h 1,h P p k we can write h ( m 4 1 ) ( m h ) h, z E. 1.6 We note that P p k P p k,andp m p P m,see 4. Definition 1.. Let f A. Then, f is said to belong to the class k R m if and only if zf /f P m p k for k,, m, and z E. For m,k, the class R R coincides with the class S of starlike functions, and R m R m consists of analytic functions with bounded radius rotation, see 5, 6.Also k R is the class ST studied by several authors, see 7, 8. Definition 1.3. Let f A. Then, f k B m α, β, h, g if and only if f is as given by 1.1 for some g k R, h P m p k in E with k,, m, α>andβ real. When m andk, we obtain the class B α, β, h, g of Bazilevic functions. We shall assume throughout, unless otherwise stated, that k,,m,α >,β real and z E.. Preliminary Results Lemma.1 see 3. Let k <, and let β,δbe any complex numbers with β / and Re β k/ k 1 δ >.Ifh is analytic in E,h 1 and satisfies zh } h p k β h δ.1 and q k is analytic solution of zq k q k } p k, β q k δ. then q k is univalent, h q k p k,.3 and q k is the best dominant of.1. Lemma. see 9. Let q be convex in E and j : E C with Re j >, z E. Ifp is analytic in E with p 1 and satisfies p j zp } q,thenp q. Lemma.3 see 9. Let u u 1 iu,v v 1 iv, and let ψ u, v be a complex-valued function satisfying the conditions i ψ u, v is continuous in a domain D C,

4 4 Applied Mathematics ii, 1 D and Re ψ 1, >, iii Re ψ i u,v 1, whenever i u,v 1 D and v 1 1/ 1 u. If h 1 c 1 z c z is a function analytic in E such that h, zh D and Re ψh,zh } > for z E,thenRe h > in E. 3. Main Results Theorem 3.1. Let 1/ 1 γ zf /f γ} P m p k,forz E and γ, 1. Define g [ z 1/α c 1 z c t c 1 f t dt] α, α >, c C, Re c. 3.1 Then, 1/ 1 γ zg /g γ} P m p k in E. In particular g k R m in E. Proof. Let zg g ( 1 γ ) p γ, 3. where p is analytic in E with p 1, and let p ( m 4 1 ) ( m p ) p. 3.3 From 3.1 and 3., we have g α [ α ( 1 γ ) p c αγ ] f α. 3.4 Logarithmic differentiation of 3.4 and some computation yield p zp α ( 1 γ ) p ( c αγ ) 1 zf } 1 γ f γ. 3.5 That is p zp α ( 1 γ ) p ( c αγ ) P ( ) m pk in E. 3.6 Let φ a,b z n zn / n 1 a b. Then, ( p φ a,b z ) p a( zp ) p b. 3.7

5 Applied Mathematics 5 Using convolution technique 3.7 with a 1/α 1 γ,b c αγ /α 1 γ, we obtain, from 3.3 and 3.6, zp i p i } α ( 1 γ ) p i ( c αγ ) p k in E, i 1,. 3.8 Since Re α 1 γ k/ k 1 c αγ}, we apply Lemma.1 with β α 1 γ,δ c αγ to obtain p i q k p k, where q k is the best dominant and is given as 1 q k β ( t β δ 1 exp tz z p k u 1 u β du) dt 1 δ β. 3.9 Consequently, p P m p k in E, and this completes the result. As a special case, we prove the following. Corollary 3.. Let k and let 1/ 1 γ 1 zf /f γ 1 } P m in E. Then, for g defined by 3.1, 1/ 1 γ zg /g γ} P m in E where γ (c αγ1 1 ) (c αγ1 1 ) 8α }. 3.1 Proof. We can write zf f ( 1 γ 1 ) h γ1, 3.11 where h P m in E. Now proceeding as before, we have, with zg g ( 1 γ ) p γ ( m 4 1 ) (1 ) γ p1 γ } ( m 4 1 ) (1 ) γ p γ } 3.1 ( 1 γ ) p γ ( 1 γ ) zp α ( 1 γ ) p ( c αγ ) zf f Using convolution technique together with 3.11,weobtain (1 ) Re γ pi ( ( ) ) 1 γ zp γ γ 1 i } α ( 1 γ ) p i ( c αγ ) >, 3.14 for i 1,.

6 6 Applied Mathematics We construct the functional ψ u, v by taking u p i,v zp i as ψ u, v ( 1 γ ) u ( γ γ 1 ) ( 1 γ ) v α ( 1 γ ) u ( c αγ ) The first two conditions of Lemma.3 are clearly satisfied. We verify condition iii as follows. Re ψ i u,v 1 ( ( ) } ) 1 γ v1 γ γ 1 Re iα ( 1 γ ) u ( c αγ ), ( ( )( ) ) c αγ 1 γ v1 γ γ 1 ( ) ( c αγ α 1 γ ), u ( ( )( )( ) ) c αγ 1 γ 1 u γ γ 1 ], A Bu C, [ (c αγ ) α ( 1 γ ) u ( v 1 1 ) u, 3.16 where A γ γ 1 c αγ 1 γ c αγ,b α γ γ 1 1 γ 1 γ c αγ,c c αγ α 1 γ u >. Re ψ iu,v 1 if and only if A,B. From A, we obtain γ as given by 3.1 and B ensures that γ, 1. Now proceeding as before, it follows from 3.1 that p P m, and this proves our result. By assigning certain permissible values to different parameters, we obtain several new and some known result. Corollary 3.3. Let f k R k ST. Then, it is known that f S γ 1,γ 1 k/ k 1 and, form Corollary 3., it follows that g S γ where γ is given by 3.1. Also a starlike function is k-uniformly convex for z <r k, 1 r k k 1, see 8. 4k 6k Therefore, for f k R, it follows that 1/ 1 γ zg /g γ} p k for z <r k,where γ is given by 3.1. As special cases we note the following. i For k, we have r 1/ 3 and f S implies that g C γ,with γ c 1 c 1 8α }. 3.18

7 Applied Mathematics 7 ii When k 1, we have γ 1 1/,γ / c α 1 1/ c α 1 8α and r 1 Theorem 3.4. Let F k B m α, β, p, f,f k R,p P m p k. Define, for Re αk/ k 1 c iβ >, G [ z 1/ α iβ c 1 z c t c 1 F t dt] α iβ Then, G k B m α, β, h, g in E, whereg is given by 3.1, and h is analytic in E with h 1. Proof. Set zg G α iβ 1 z iβ g α h ( m 4 1 ) ( m h ) h. 3. We note that h is analytic in E with h 1. From 3., we have [ ]} z iβ g α zh h α zg c iβ zf F α iβ 1, g 3.1 using 3.1, wenotethat ( ) f α α zg c iβ. g g 3. From 3.1 and 3., it follows that } zh h αh ( c iβ ) ( ) P m pk, 3.3 where h zg /g P p k since g k R by Theorem 3.1. It can easily be seen that g S k/ k 1 and Reαzg /g c iβ} >. Now, using 3.8, we can easily derive hi j ( zh i )} p k in E, i 1,, 3.4 where 1/j αzg /g c iβ} and Re j >. Applying Lemma., it follows from 3.4 h i p k in E and therefore h P m p k in E. This completes the proof.

8 8 Applied Mathematics Theorem 3.5. Let f be given by 1.1 with h 1, α β 1/ e iγ zg /g } P m p k α β 1/ e iγ α iβ, γ <π/. Then, for z E i e iγ zf /f cos γ p i sin γ, p P m p k, ii For α iβ t α iβ, t 1, k B m ( α, β, 1,g ) kt Bm ( α,β, 1,g ). 3.5 Proof. i From 1.1, we have 1 zf f ( α 1 iβ )zf ( ) f α zg iβ H, g ( ) H P m pk in E. 3.6 Define a function p analytic in E by e iγ zf f cos γ( p ) i sin γ, γ tan 1 β α. 3.7 We can easily check that p 1. Now, from 3.6 and 3.7, we have [ zp p i tan γ ] ( ) αp iβ P m pk in E. 3.8 That is [ αzp ] ( ) αp iβ P m pk, αp iβ 3.9 and, with h αp iβ m/4 1/ h 1 m/4 1/ h, we apply convolution technique used before to have h i zh } i p k in E. 3.3 h i Applying Lemma, it follows that h i q k p k, z E, 3.31 where q k is the best dominant and is given by [ 1 q k exp ( tz 1 p k u 1 du)]. 3.3 u From 3.31, we have h αp iβ P m p k in E, and this proves part i.

9 Applied Mathematics 9 ii From part i, we have (α β ) 1/ e iγ zf f H 1, H 1 P m ( pk ) in E Now, 1 zf f ( α 1 iβ )zf f 1 zf f ( α 1 iβ )zf } f t 1 (α β ) 1/ e iγ zf f, ( ) H t 1 H 1, H i P m pk, i 1,, [( t 1 1 ) H 1 1 ] t t H, 3.34 th, t 1, H P m p k,sincep m p k is convex set, see 8. Therefore, f kt B m α,β, 1,g for z E. This completes the proof. As a special case, with m,k, we obtain a result proved in 1. By assigning certain permissible values to the parameters α, β and m, we have several other new results. Acknowledgment The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities and environment References 1 I. E. Bazilevič, On a case of integrability in quadratures of the Loewner-Kufarev equation, vol , pp , W. Kaplan, Close-to-convex schlicht functions, The Michigan Mathematical Journal, vol. 1, pp , S. Kanas, Techniques of the differential subordination for domains bounded by conic sections, International Mathematics and Mathematical Sciences, no. 38, pp , 3. 4 B. Pinchuk, Functions of bounded boundary rotation, Israel Mathematics, vol. 1, pp. 6 16, A. W. Goodman, Univalent Functions, vol. 1 and, Mariner Publishing, Tampa, Fla, USA, K. Inayat Noor, Some properties of analytic functions with bounded radius rotation, Complex Variables and Elliptic Equations, vol. 54, no. 9, pp , 9. 7 S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, Computational and Applied Mathematics, vol. 15, no. 1-, pp , K. I. Noor, On a generalization of uniformly convex and related functions, Computers & Mathematics with Applications, vol. 61, no. 1, pp , 11.

10 1 Applied Mathematics 9 S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 5 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA,. 1 P. J. Eenigenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade, On a subclass of Bazilevič functions, Proceedings of the American Mathematical Society, vol. 45, pp. 88 9, 1974.

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