Generalized Class of Sakaguchi Functions in Conic Region

Transcription

1 Iteratioal Joural of Egieerig ad Techical Research (IJETR) ISSN: , Volume-3, Issue-3, March 05 Geeralied Class of Saaguchi Fuctios i Coic Regio Saritha. G. P, Fuad. S. Al Sarari, S. Latha Abstract I this paper, we use the cocept of Saaguchi type fuctios, Jaowsi fuctios ad the coic regios are combied to defie a class of fuctios i a ew iterestig coic domai. We prove coefficiet iequalities ad iclusio results. Idex Terms Aalytic fuctios, Saaguchi type fuctios, Coic domais, Jaowsi fuctios, Starlie fuctios, Uiformly covex fuctios. 00 AMS Subect Classificatio: 30C45 I. INTRODUCTION Let A be the class of fuctios of form f a, () Which are aalytic i the ope uit dis U : ad, with ormaliatio f 0 0 ad f 0. Cosider the coic regio, * Correspodig author give by u iv : u u v. This domai represets the right half plae for 0, hyperbola for 0, a parabola for ad ellipse for. The fuctios p fuctios for these coic regios play the role of exteral Mauscript received March 4, 05. Saritha. G. P, Departmet of Mathematics, Bahubali College of Egieerig, Shravabelagola, Hassa 57335, Idia Fuad. S. Al Sarari, Departmet of Studies i Mathematics, Uiversity of Mysore, Maasagagothri, Mysore , Idia S. Latha, Dertmet of Mathematics, Yuvaraa s College, Uiversity of Mysore, Mysore , Idia. p, 0 log,. sih arccos arcta h, 0. u si t dx,, 0 R t x tx () u, t 0,, t U ad is chose tx Where such that R t cosh, R t 4Rt complete elliptic itegral of the first id ad complemetary itegral. 8 arccos,0 8, 4 is the Legedre s R t is Rt p... 9 t t R t,. (3) Usig the cocept of coic regios we defie the followig: Defiitio.. A fuctio f UB,,, s, t, A is said to be the class for 0, 0,0,,0 s, t with s t if ad oly if J,,, s, t, f J,,, s, t, f, 408

2 Geeralied Class of Saaguchi Fuctios I Coic Regio s t f s t f J,,, s, t, f. f s f t f s f t Or equivaletly p is defied by (6). J,,, s, t, f p, This class geeralies various classes studied by Khalida Iayat Noor ad Sarfra Naea Mali i [6], Kaas ad Wisiowsa [3, 0], Shams ad Kulari [4], Kaas [7], Mocau [], Goodma [5]. II. MAIN RESULTS Theorem.. A fuctio f A ad of the form () is i the class UB,,, s, t, if it satisfies the coditio ;,,, s, t, (4) Where ;,,,,,., s t u s t u a u s, t u s, t u s,,, u s t u a u s, ta a were 0, 0,0,,0 ad, u s t s t Proof. Assumig equatio (8) holds, the it suffices to show that Now cosider 0 u s,, J,,, s, t, f J,,, s, t, f. J,,, s, t, f, the s t f s t f f s f t f s f t s t f f s f t f s f t f s f t f s f t f s f t s t f f s f t 5 Now from () ad we get 409

3 Iteratioal Joural of Egieerig ad Techical Research (IJETR) ISSN: , Volume-3, Issue-3, March 05,,,,,, s tf f s f t s t a u a0 u0 s t a u s t s u s, 0 0 s t u s, a , u a s t u s, a s t u s, ta u s, a. Similarly, we get us, ta ad f s f t f s f t s t u s, t. u s, ta a f s f t f s f t s t u s, u s, a, Usig the above equalities i (9), we get u s, t u s, t u s, u,,., u s t u a,, u s t a a u u, a, u s t a a a u s t u a,,.,,,,., u s t a a a u s t u a,,, u s t u s t u u u s t u s a a,,., t,,,. u s t a a u u s t a a N,, u s t u s t a a, u u, u a,., t s t u s a a,,., t Sice u u s t u s a a J,,, s, t, f J,,, s, t, f J,,, s, t, f, (6) 40

4 Geeralied Class of Saaguchi Fuctios I Coic Regio The u u s t u s a a u s, t u s, t u s,,,., t,, t u s t u s a a,,., t u u s t u s a a, t u s a a,,., t The last expressio is bouded by if u u s t u s a a,,, u s t u s t u u s, t u s, ta a u s, t a a u s, u s, t. u s, a This completes the proof. Whe s, t o, we have the followig result, proved by Khalida Iayat Noor ad Sarfra Nawa Mali i [6]. Corollary.. A fuctio f A ad from () i the class,,, for,, 0, 0 if it satisfies the coditio ;,,, (7) ;,, 0 a a a gg a a a. For s, t 0, 0, we have followig result due to Shams ad Kulari [4]. Corollary.3. A fuctio f A ad from () i the class SD,, if it satisfies the coditio a. a a a a a a 4

5 This implies that Iteratioal Joural of Egieerig ad Techical Research (IJETR) ISSN: , Volume-3, Issue-3, March 05 a For s, 0, we arrive at Shams ad Kulari et result i [4]. Corollary.4. A fuctio f A ad from () i the class KD,, if it satisfies the coditio a. a a a a a a This implies that a Also for s, t 0, gb 0, 0 the we get the well-ow Kaas s result [7]. Corollary.5. A fuctio f A ad from () i the class UM,, if it satisfies the coditio ;, ; a a a a a a. For s, t 0, 0, 0, the we get result proved by Kaas ad Wisiowsa i [3]. Corollary.6. A fuctio f A ad from () i the class ST, if it satisfies the coditio a. Also for s, t 0, 0, 0, the we have the followig ow result, proved by Silverma i [8] Corollary.7. A fuctio f S *, if it satisfies the coditio A ad from () i the class a. REFERENCES [] P. T. Mocau, Ue propriete de coveite geerlise das la theorie de la represetatio coforme, Mathematica (Clu) (969) []. R. Sigh ad M. Tygel, O some uivalet fuctios i the uit disc, Idia. J. Pure. Appl. Math. (98) [3] 3. S. Kaas, A. Wisiowsa, Coic domais ad starlie fuctios, Roumaie Math. Pures Appl. 45(000) [4] 4. S. Shams, S. R. Kulari, J.M. Jahagiri, Classes of uiformly starlie ad covex fuctios, It. J. Math. Math. Sci. 55(004) [5] 5. A. W Goodma, O uiformly covex fuctios, A. Polo. Math. 56 (99) [6] 6. K.L. Noor ad S. N. Mali, O geeralied bouded Mocau vaiatio associated with coic domi, Math. Comput. Model. 55 (0) [7] 7. S. Kaas, Alterative characteriatio of the class UCV ad related classes of uivalet fuctios Serdica Math. J. 5 (999) [8] 8. H. Selverma, Uivalet fuctios with egative coefficiets, Poroc. Amer. Math.Soc. 5 (975) [9] 9. S. Kaas, Coefficiet estimates i subclasses of the Caratheodory class related to coical domais, Acta. Math. Appl. Acta. Math. Uiv. Comeia 74 () (005)49-6. [0] S. Kaas, A. Wisiowsa, Coic regios ad -uiform covexity, J. comput. Appl. Math. 05 (999)