Coefficient Inequality For Some Subclass Of Meromorphic Functions

Transcription

1 dvaces i Theoretical ad pplied Mathematics IN Volume Numer 3 6 pp Research Idia Pulicatios oefficiet Iequality or ome uclass Of Meromorphic uctios Karthiyayii. O ad ivasaari. V Departmet of ciece ad Humaities PEIT-aalore outh ampus m efore Electroic ity Hosur Road aalore-56 INDI arthiyayii. roy@pes. edu sivasaariv@pes. edu stract I this paper we defie two ew suclass ad of meromorphic fuctios associated with a liear operator. We ive ufficiet coditios for a fuctio to e i these suclasses. oectios to earlier ow results are also idicated. Keywords ad Phrases: alytic fuctios Meromorphic fuctios Meromorphic starlie fuctios Meromorphic covex fuctios. Mathematics uject lassificatio: Primary 345. Itroductio Let deote the class of fuctios of the form. which are aalytic ad meromorphic uivalet i the puctured uit disc {: : } Motivated y the liear operator defied y ateh. i et al. i [3] we defie for i the form. the followi...

2 76 Karthiyayii. O ad ivasaari. V ] [ ad [ ] ] [ hece it ca e easily see that... ] [ N. I this paper we itroduce the suclasses of meromorphic fuctio ad associated with the liear operator ad coefficiet iequalities for fuctios i these suclasses are otaied. Defiitio. : or the real costats ad ; - let cosists of fuctios which satisfy the followi coditio:... ad p. 3 Defiitio. : or the real costats ad ; - let cosists of fuctios which satisfy the followi coditio:... ad p. 4 Remar. 3. y ivi special values for we otai the followi suclasses of meromorphic fuctios that were studied earlier i the literature r r ad T ad ad The class ad were studied y Raha W. Irahim ad Masila Darus [6]. The class ad were studied y. Latha ad L. hivarudrappa [8]. The classes T r ad r were studied y hieyoshi Owa ad Nicolae N. Pascu [7].

3 oefficiet Iequality or ome uclass Of Meromorphic uctios 77. Mai Results We prove ufficiet coditios for to e i the classes ad. Theorem. : Let. If ] [ ] [. holds ad > the. Proof: ssume that ad satisfies.. It is sufficiet to show that that is ' osider '. ] [ ] [ Hece. is ouded y if ] [ Equivaletly

4 78 Karthiyayii. O ad ivasaari. V ] [ ] [ ] [ where >. This completes the proof. Theorem. : Let. If ] [ ] [. 3 holds ad > the. Proof: ssume that ad statisfies. 3. It is sufficiet to show that that is ' osider '. 4 ] [ ] [ Hece. 4 is ouded y if ] [ ] [ where >. This completes the proof. Remar. 3.: y specialii o the parameters as i Remar. i

5 oefficiet Iequality or ome uclass Of Meromorphic uctios 79 theorem. ad theorem. we otai correspodi theorem. ad theorem. i [6] [8] [7]. Refereces [] Dure. P. L 97 oefficiets of meromorphic chlicht fuctios Proc. mer. Math. oc. 8 pp [] Dure. P. L 983 ivalet uctios Grudlehre der Mathematische Wis- seschafte prier New Yor 59. [3] ateh. i ad dul Rahma. Juma 4 Estimati oefficiets for su classes of meromorphic i-uivalet fuctios associated with liear operator TWM J. pp. E. Math 4 pp [4] M. Darus ad R. W. Irahim 8 oefficiet iequalities for a ew class of uivalet fuctios Loachevsii J. Math. 9 pp. -9. [5] Naeu cho ad hieyoshi Owa 4 O partial sums of certai meromorphic fuctios Joural of iequalities i pure ad applied mathematics 5 rticle 3. [6] R. W. Irahim ad M. Darus Partial sums for certai classes of meromorphic fuctios Tama J. Math. 4 pp [7] hieyoshi Owa ad Nicolae. Pascu 3 oefficiet Iequalities for certai classes of meromorphically starlie ad meromorphically covex fuctios Joural of Iequalities i Pure ad pplied mathematics 4 rticle 7. [8]. Latha ad L. hivarudrappa 6 Partial sum of meromorphic fuctios J. Ieq. Pure. ppl. Math. 74 rticle 4 pp. -8.

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