Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: 197-537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics, Uiversity of Mysore, Maasagagotri, Mysore 57 6, Idia Departet of Matheatics, Yuvaraja s Collega, Uiversity of Mysore, Mysore 57 5, Idia Copyright c 1 Sarari ad Latha. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Abstract. The object of the preset paper is to derive the itegral represetatio for classes ivolvig the otio of (,-syetrical fuctios with bouded boudary rotatio ad bouded radius rotatio. Soe ore properties like radius of uivalet ad starlike are also ivestigated. Keywords: covex fuctios, starlike fuctios, fuctios of bouded boudary rotatio, bouded radius rotatio, (,-syetric poits. 1 AMS Subject Classificatio: 3C5. 1. Itroductio-preliiaries Let A deote the class of fuctios of for (1 f (z = z + = a z, which are aalytic i the ope uit disk U = z : z C ad z < 1. Let S deote the subclass of A cosistig of all fuctios which are uivalet i U. We also deote by S,K the failiar subclasses of it cosistig of fuctios which are respectively starlike ad covex i Correspodig author Received March 3, 1 9
ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 95 U. It is kow that f (z S if ad oly if z f (z = z exp log( it d(t, for soe (t M. Pichuk [1] geeralized the class S by allowig (t to rage over the class M k. More precisely a fuctio f (z is said to be i the class U k if f (z = z exp z log( it d(t, (t M k i.e, (t is a real valued fuctio of bouded variatio o [,π] satisfyig the coditios. ( d(t =, d(t k. Geoetrically the coditio is that the total variatio of the agle which the radius vector f (re iθ akes whit the positive real axis is bouded above by πk as z describes the circle z = r for z < 1. Thus U k the class of fuctios with radius rotatio bouded by πk. Siilarly V k deotes the class of fuctios f defied o U which ap coforally oto a iage doai of boudary rotatio at ost kπ. Hece f (z V k, if ad oly if f (z = exp log( it d(t, (t M k. It is easy to see that U is the class of starlike fuctios ad V is the class of covex fuctios. Let P k deote the class of fuctios which are aalytic i U ad have the represetatio (3 p(z = 1 1 + ze it d(t, it where (t M k. Clearly we have p = p ad f U k ad V k if ad oly if z f f ad 1 + z f f belog to P k. For p P k, the it has the followig properties (1 p( = 1, ( Rp(z dθ kπ, where k ad z = re iθ, r < 1. Liczberski ad Polubiki [] itroduce the otio of (,-syetrical fuctios ( = 1,,3,..., =,1,.., 1 which is geeralizatio of otios of eve odd ad syetrical fuctios. They also geeralized the kow result that each fuctio defied i syetrical subset ca be u- iquely represeted as the su of a eve fuctio ad odd fuctio.
96 FUAD. S. M. AL SARARI, S. LATHA Defiitio 1.1. Let ε = (e πi ad =,1,,.., 1 where is a atural uber. A fuctio f : U C is called (, -syetrical if f (εz = ε f (z, z U. The faily of all (,-syetrical fuctios is deoted be S (,. S (,, S (1, ad S (1, are respectively the classes of eve, odd ad -syetric fuctios. We have the followig decopositio theore. Theore 1.. [] For every appig f : U C, there exists exactly the sequece of (,- syetrical fuctios f,, where f (z = f, (z, = ( f, (z = 1 ε v f (ε v z. v= ( f A ; = 1,,...; =,1,,..., 1. The followig idetities follow directly fro ( (5 f,(z = 1 ε v v f (ε v z, f,(z = 1 v= ε v v f (ε v z, v= (6 f, (ε v z = ε v f, (z, f,(ε v z = ε v v f,(z. Defiitio 1.3. Let U k (, deote the class of fuctios f A satisfies f ( =, f ( = 1 ad, where f, (z is defied by (. z f (z f, (z P k, Defiitio 1.. Let V k (, deote the class of fuctios f A satisfies f ( =, f ( = 1 ad where f, (z is defied by (. (z f (z f,(z P k,
ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 97 Reark 1.5. f V k (, if ad oly if z f U k (,. Spacial cases (i For k = 1, = 1 we get Sigh ad Tygel i [8]. (iifor = = 1 we get paatero i []. (iiifor k =, = 1, = we get Sakaguchi i [13]. I our paper, we also eed the the followig leas. Lea 1.6. [3] Suppose p(z P k. The zp (z R r(k r + kr p(z (1 r (1 kr + r, wher z = r,k ad z < R = k k. For k, zp (z R kr + (8 k + k r kr 3 p(z (1 r (1 kr + r. The above iequality is sharp for fuctio p(z = 1 kz+z 1 z.. Mai results Theore.1. A fuctio f A belogs to U k (,, the (7 f, (z = z exp 1 log( v= where f, (z is defied by ( ad (t is defied (. Proof. Suppose that f U k (,. It follows that πv i(t d(t. (8 z f (z f, (z = p (z, where (9 p (z = 1 Substitutig z by ε v z i (8 respectively 1 + ze it it d(t. (1 zε v f (ε v z f, (ε v z = p (ε v z.
98 FUAD. S. M. AL SARARI, S. LATHA The (11 zε v v f (εz v = 1 f, (z 1 + zε v e it 1 zε v e it d(t, or (1 zε v v f (εz v = 1 f, (z 1 + ze πv i(t πv i(t Let (v =,1,,... 1 i (1 ad suig the we get (13 by itegral (13 we have (1 f,(z f, (z 1 z = 1 1 + ze z v= πv i(t πv i(t ( f, (z log = 1 z log[ v= fro (1 we get (7. Hece the proof is coplete. Theore.. A fuctio f A belogs to U k (,, the [ ] (15 f (z = 1 z exp 1 v= log(1 ye i(t πv where f, (z is defied by ( ad (t is defied (. Proof. Suppose that f U k (,. It follows that d(t d(t. d(t 1 z, πv i(t. ]d(t, 1 + ye it 1 ye it d(t dy (16 z f (z f, (z = p (z. The By usig Theore.1, we get (17 f (z = exp 1 v= z f (z = f, (zp (z. log( i(t πv fro (17 we get (15. Hece the proof is coplete. d(t. 1 1 + ze it it Corollary.3. For = 1 ad = 1 i Theore.1 we get Paatero []. d(t, By usig the sae ethod i Theore.1, we have the followig corollaries.
ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 99 Corollary.. A fuctio f A belogs to V k (,, the f,(z = exp 1 (18 log( v= where f, (z is defied by ( ad (t is defied (. πv i(t Corollary.5. A fuctio f A belogs to V k (,. The [ ] f (z = 1 z exp 1 i(t πv log(1 ye π (19 d(t z v= where f, (z is defied by ( ad (t is defied (. d(t 1 + ye it 1 ye it Theore.6. A fuctio f A belogs to U k (,. The f, (z i U k. Proof. Suppose that f U k (,. It follows that, d(t dy, ( z f (z f, (z = p (z. Substitutig z by ε v z i ( respectively (1 zε v f (ε v z f, (ε v z = p (ε v z. Now let (v =,1,,... 1 i (1 ad suig the we get ( z f,(z f, (z = 1 1 p (εz. v v= It is vivid that 1 1 v= p (ε v z be bogs to P k. Hece the proof is coplete. Theore.7. Let f U k (, ad let F(z = z f (z. The F(z is starlike for z < r, where r is the least positive root of the equatio 1 3kr + (k + 6r 3kr 3 + r =, where z = r ad k. For k, the F(z is starlike for z < r 3 where r 3 is the least positive root of the equatio 6kr + (1 k + 3k r kr 3 + r =. However the boud r 3 is ot sharp whe k <.
5 FUAD. S. M. AL SARARI, S. LATHA Proof. Let f U k (,. The It follows that (3 or ( Hece (5 Therefore, we have ad The F(z = zexp 1 v= zf (z F(z = 1 + 1 v= R R zf (z F(z = 1 v= zf (z 1 = R F(z log( ze v= πv i(t πv i(t πv i(t p (ε v z + zp (z p (z. p (ε v z d(t d(t + zp (z p (z, + R zp (z. p (z.p (z. zp (z r(k r + kr p (z (1 r (1 kr + r, where z = r,k, R 1 R v= p (ε v z where z = r < R = k k. Hece R 1 kr + r (1 r, where z = r,k. zf (z 1 kr + r F(z (1 r + r(k r + kr (1 r (1 kr + r (1 kr + r r(k r + kr (1 r (1 kr + r, provided Q(r = 1 3kr + (k + 6r zf (z F(z 3kr 3 + r >. The equatio Q(r = has a uique positive root i (,R. For k, by usig (5, we have zf (z 1 kr + r R F(z (1 r + kr + (8 k + k r kr 3 (1 r (1 kr + r, where z = r < R = k k. Hece R > provided Also D(r = has a root i (,R. zf (z F(z D(r = 6kr + (1 k + 3k r kr 3 + r >.
ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 51 Corollary.8. Let f U k (,. The f is covex for z < r, where r is the least positive root of the equatio 1 3kr + (k + 6r 3kr 3 + r =, where z = r ad k. For k, the f (z is covex for z < r 3, where r 3 is the least positive root of the equatio 6kr + (1 k + 3k r kr 3 + r =. However the boud r 3 is ot sharp whe k <. Theore.9. Let f U k (, ad let F(z = z f, (t f, ( t 1 t dt. The F(z is i V k. Proof. Sice f U k (,, we have ad The or (zf (z F (z (zf (z F (z f, (z = z exp F (z = f,(z f, ( z 1 z = 1 + 1 v= = 1 1 v= = 1 1 ze 1 + ze v= i(t πv i(t πv i(t πv i(t πv log( d(t 1 d(t + 1, πv i(t v= v= d(t ze 1 + ze 1 + ze 1 p (εz v + 1 v= p ( εz v. v=. i(t πv i(t πv i(t πv i(t πv d(t d(t Sice p (z P k so 1 v= p (ε v z also i P k, by settig q(z = 1 v= p (ε v z, we have (zf (z F = 1 (z q(z + q( z where q(z P k, = 1 ( ( k + k q 1 (z q (z + 1 ( ( k + k q 1 ( z q ( z (zf (z F (z ( k + q1 (z + q 1 ( z = ( k q (z + q ( z,
5 FUAD. S. M. AL SARARI, S. LATHA where q i (z P, i = 1,, also q i(z+q i ( z P, i = 1,. Hece (zf (z ( ( k + k F = w 1 (z w (z, (z where w i (z P, i = 1,. Hece which eas F(z V k. (zf (z F (z P k, Coflict of Iterests The authors declare that there is o coflict of iterests. REFERENCES [1] B. Pichuk, Fuctios of bouded boudary rotatio, Isreal J. Math. 1 (1971, 6-16. [] V. Paatero, Uber Gebiete vo besehrukter Raddrehug, A. Aced. Sci. Fe. Ser A. 37 (1933, 9. [3] K. S. Padaabha ad R. Parvatha, O fuctios with bouded boudary rotatio, Idia J. pure appl. Math. 6 (1975, 136-17. [] P. Liczberski, J. Polubiski, O ( j,k-sytrical fuctios, Matheatica Boheica 1 (1995, 13-5. [5] P. L. Dure, Uivalet Fuctios, Spriger-Verlag (1983. [6] K. I. Noor, O quasi-covex fuctios ad related topics, It. j. Math. Math. Sci. 1 (1987, 1-58. [7] R. M. Goel ad B. S. Mehrok, Soe ivariace properties of a subclass of close-to-covex fuctios, Idia J. Pure Appl. Math. 1 (1981, 1-19. [8] R. Sigh ad M. Tygel, O soe uivalet fuctios i the uit disc, Idia. J. Pure. Appl. Math. 1 (1981, 513-5. [9] Z. G. Wag, C. Y. Gao ad S. M. Yua, O certai subclasses of close-to-covex ad quasi-covex fuctios with respect tok-syetric poits, J. Math. Aal. Appl. 3 (6, 97-16. [1] S. Owa, M. Nuokawa, H. Saitoh ad H. M. srivastava, Close-to-covexity,starlikess,ad covexity of certai aalytic fuctios, Appl. Math. Lett. 15 (, 63-69. [11] V. Ravichadra, Starlike ad covex fuctios with respect to cojugate poits, Acta Math. Acad. Paedagog. Nyhazi. (, 31-37. [1] T. N. Shauga, O α-quasi-covex fuctios, Idia J. Pure Appl. Math. (1989, 915-9. [13] K. Sakaguchi O certai uivalet appig, J. Math. Soc. Japa. 11 (1959, 7-75.