AN APPLICATION OF q-calculus TO HARMONIC UNIVALENT FUNCTIONS (Suatu Penggunaan q-kalkulus untuk Fungsi Univalen Harmonik)

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1 Journal of Qualty Measurement and Analyss JQMA 14(1) 018, Jurnal Penguuran Kualt dan Analss AN APPLICATION OF -CALCULUS TO ARMONIC UNIVALENT FUNCTIONS (Suatu Penggunaan -Kalulus untu Fungs Unvalen armon) SAURAB PORWAL 1 & ANKITA GUPTA ABSTRACT The purpose of the present paper s to ntroduce a new subclass of harmonc unvalent functons assocated wth -calculus. We obtan coeffcent condtons, dstorton bounds, extreme ponts, convoluton condton and convex combnaton for functons belongng to ths class. We also dscuss a class-preservng ntegral operator and -Jacson type ntegral operator for ths class of functons. Keywords: harmonc; unvalent functon; -calculus ABSTRAK Maalah n bertujuan memperenalan suatu subelas fungs unvalen harmon baharu yang datan dengan -alulus. Dperoleh batas-batas peal, batas erotan, tt estrem, syarat onvolus dan gabungan cembung untu fungs yang terandung dalam elas n. Juga dbncangan elas mengeal pengoperas pengamr dan pengoperas pengamr jens -Jacson untu elas fungs tersebut. Keywords: harmon; fungs unvalen; -alulus 1. Introducton A contnuous complex-valued functon f u v s sad to be harmonc n a smply-connected doman D f both u and v are real harmonc n D. In any smply-connected doman we can wrte f h g, where h and g are analytc n D. We call h the analytc part and g the co-analytc part of f. A necessary and suffcent condton for f to be locally unvalent and sense-preservng n D s that h( ) g( ), D, see Clune and Shel-Small (1984) and Duren (004). Let S denote the class of functons f h g that are harmonc unvalent and sense-preservng n the open unt dsc U : 1 for whch f(0) f (0) 1 0. Then for f h g S we may express the analytc functons h and g as, 1 h( ) a, g( ) b, b 1. (1) Note that S reduces to class S of normaled analytc unvalent functons f the co-analytc part of ts member s ero Now, we recall the concept of -calculus whch may be found n Aral et al. (013). For N, the -number s defned as follows: 1 [ ], ()

2 Saurabh Porwal & Anta Gupta ence, [ ] can be expressed as a geometrc seres 1, when, then the seres 1 converges to. As 1, [ ] and ths s the boomar of a -analogue the lmt as 1 1 recovers the classcal object. The -dervatve of a functon f s defned by f ( ) f ( ) D ( f ( )), 1, 0, and D ( f (0)) f (0) provded f (0) exsts. ( 1) For a functon h(), we observe that 1 D h D ( ( )) ( ) [ ]. 0 Then lm D h lm h ( ( )) 1[ ] ( ), where h' s the ordnary dervatve. The -Jacson defnte ntegral of the functon f s defned by 0 n n f ( t) d t (1 ) f ( ),. n0 In 1994 Uralegadd et al. (1994, 1995) ntroduced the analogues subclasses of starle, convex and close-to-convex analytc functons wth postve coeffcents and opened up a new and nterestng drecton of research n the theory of analytc unvalent functons. Motvated wth the ntal wor of Uralegadd et al. (1994, 1995) many researchers (e.g. Dxt and Chandra (008), Dxt and Patha (003), Dxt et al. (013), Porwal and Dxt (010) and Porwal et al. (011)) ntroduced and studed varous new subclasses of analytc unvalent functons wth postve coeffcents. In 010, Dxt and Porwal (010) ntroduced a new subclass of harmonc unvalent functon wth postve coeffcents and opened up a new drecton of research n the theory of harmonc unvalent functons. After the appearence of ths paper, several researchers (e.g. Patha et al. (01), Porwal and Aouf (013), Porwal et al. (01)) generalsed the result of Dxt and Porwal (010). Porwal and Dxt (013) nvestgated new subclasses of harmonc starle and convex functons. These results were generalsed n Porwal 015a, Porwal 015b and 01. In the present paper, analogues to the above mentoned wors, we ntroduce a new subclass of harmonc unvalent functons by usng -calculus. Let [, ] the followng condton M denote the famly of harmonc functons of the form f h g satsfyng ( Dh( )) ( Dg( )) 4 for 1, 0 1. (3) h( ) g( ) 3 Further, let V denote the subclass of S consstng of functons of the form, (4) 1 f ( ) a b. Also, we defne V (, ) M (, ) V. 8

3 An applcaton of -calculus to harmonc unvalent functons In ths study, we obtan coeffcent bound, extreme ponts, dstorton bounds, convoluton, convex combnaton for functons n the class V (, ). Fnally we dscuss a class preservng ntegral operator.. Man Results Theorem.1. Let the functon f h g be gven by (1). If [ ] [ ] b a 1, (5) 4 then f M [, ], where 1 and 0 < < 1. 3 Proof. Let [ ] [ ] a 1 It suffces to show that, ( Dh( )) ( Dg( )) 1 h( ) g( ) ( Dh( )) ( Dg( )) ( 1) h( ) g( ) 1 [ ] [ ] 1 a b 1 [ ] [ ] 1 a b 1 a b 1 a b ( 1) 1 1 ( [ ] 1) a ( [ ] 1) b 1 ( 1) ( [ ] 1) a ( [ ] 1) b The last expresson s bounded above by 1, f 83

4 Saurabh Porwal & Anta Gupta ( [ ] 1) a ( [ ] 1) b ( 1) ( [ ] 1) a ( [ ] 1) b, 1 1 ( [ ] 1) a ( [ ] 1) a ( [ ] 1 [ ] 1) b ( 1), euvalent to ence ( [ ] ) a ( [ ] b ( 1), 1 [ ] [ ] b a 1 ( Dh( )) ( Dg( )) 1 h( ) g( ) ( Dh( )) ( Dg( )) ( 1) h( ) g( ). 1, U. Ths completes the proof of Theorem.1. Theorem.. A functon of the form (4) s n V (, ), f and only f, [ ] [ ] a b (6) Proof. Snce V(, ) M (, ), the f part follows from Theorem.1. For only f part we show that f V (, ) f the above condton does not hold. Note that a neccessary and suffcent condton for f h g gven by (4) s n V (, ), f, ( Dh( )) ( Dg( )), h( ) g( ) s euvalent to ( 1) ( [ ] ) a ( [ ] ) b 1 0. a b 1 The above condton must hold for all values of, r 1, upon choosng the values of on the postve real axs where 0 r 1, we must have 84

5 An applcaton of -calculus to harmonc unvalent functons 1 1 ( 1) ( [ ] ) a r ( [ ] ) b r 1 0. (7) a r b r 1 If the condton (6) does not hold then the numerator of (7) s negatve for r suffcently close to 1. Thus there exst a 0 r0 n (0, 1) for whch the uotent n (7) s negatve. Ths contradcts wth the reured condton for f V (, ) and so the proof s completed. Next, we determne the extreme ponts of the closed convex hulls of V (, ), denoted by clco V (, ). Theorem.3. A functon f clcov (, ), f and only f (8) f ( ) x h ( ) y g ( ), 1 1 where h1 ( ), h ( ), (,3,) [ ] ( = 1,,...), 1 ( x y ) 1, x 0, y 0. and 1 g (), [ ] In partcular the extreme ponts of (, ) V are h and g. Proof. Suppose that f ( ) x h ( ) y g ( ) 1 1 x y [ ] [ ]. 1 Then, [ ] 1 [ ] 1 x y 1 [ ] 1 1 [ ] and so f clcov [, ]. Conversely f f clcov [, ], set [ ] [ ] x a,,3, 4, and y b, 1,,3, 1 1 x y x , 85

6 Saurabh Porwal & Anta Gupta from Theorem. we have, 0 x 1, and 0 y 1,( 1,,3, ). We defne and by Theorem., x1 0. x x y as reured. 1 Conseuently, we obtan f ( ) x h ( ) y g ( ) Theorem.4. Let f V (, ). Then for r 1, we have, 1 1 f ( ) (1 b1 ) r b1 r and 1 1 f ( ) (1 b1 ) r b1 r. Proof. We only prove the rght hand neualty. The proof for left hand neualty s smlar and wll be omtted. Let f V (, ), tang the absolute value of f, we have, f ( ) (1 b1 ) r ( a ) b r, 1 (1 b ) r ( a b ) r, (1 b1 ) r r ( a b ), 1 (1 b1 ) r r ( a b ), 1 1 [ ] (1 b1 ) r r ( a b ), 1 1 [ ] [ ] (1 b1 ) r r a b, (1 b1 ) r r 1 b1, (1 b1 ) r r b1. Thus the proof of Theorem.4 s establshed. 86

7 An applcaton of -calculus to harmonc unvalent functons Theorem.5. Let f V (, ) and F V (, ). Then f * F V(, ) V(, ), for and Proof. Let be n V (, ). be n V (, ) 1 f ( ) a b Then the convoluton f * F s gven by and 1 F( ) A B f * F f * F 1 a A b B. We wsh to show that the coeffcents of f * F satsfy the reured condton n Theorem. for F( ) V (, ). We note that A 1 and B 1. Now for the convoluton functon f * F, we obtan, [ ] [ ] a A b B [ ] [ ] b 1 1 a 1 1, snce f ( ) V (, ). Therefore f * F V(, ) V(, ). Thus the proof of Theorem.5 s establshed. Theorem.6. The class V (, ) s closed under convex combnaton. Proof. Let f ( ) V (, ), 1,,3,, where f () s gven by 1 f ( ) a b. Then, by Theorem., [ ] [ ] b a 1 (9) The convex combnaton of f may be wrtten as t f ( ) t a t b, for Then by Theorem., we have, 1 t 1, 0 t 1. 87

8 Saurabh Porwal & Anta Gupta [ ] [ ] t a t b [ ] [ ] t a b t 1. Ths s the condton reured by Theorem., and hence we have The proof of Theorem.6 s completed. t f ( ) V (, ) A Famly of Class Preservng Integral Operator Let f() be defned by (1). Then let us defne F() by the relaton, c1 c1 F t h t dt t g t dt c c1 c1 ( ) ( ) ( ), ( 1). c 0 c 0 (10) Theorem 3.1. Let f ( ) h( ) g( ) S be gven by (4) and f V (, ) 4 1. Then F() defned by (10) s also n the class V (, ). 3 Proof. Let be n (, ). 1 f ( ) a b [ ] [ ] b a 1, where V Then by Theorem., we have, From the euaton (10) of F(), t follows that, Now, c1 c1 F( ) a b. c c 1 [ ] c1 [ ] c1 b 1 c 1 c [ ] [ ] a b 1 1 a Thus F( ) V (, ). The proof of Theorem 3.1 s completed. 88

9 An applcaton of -calculus to harmonc unvalent functons Defnton 3.1. Let f h g be defned by (1). Then the -Jacson-type ntegral operator F : s defned by the relaton, c1 c1 c1 c1 c 0 c, 0 F ( ) t h( t) dt t g( t) dt, ( c 1) (11) where [ a ] s the -number defned by () and s the class of functons of the form (1) whch are harmonc n U. Theorem 3.. Let f ( ) h( ) g( ) be gven by (4) and f V (, ) where 4 1, 0 1. Then F () defned by (11) s also n the class V (, ). 3 Proof. Let have be n V (, ) 1 f ( ) a b [ ] [ ] b a 1. Then by Theorem., we From the representaton (11) of F (), t follows that, Snce [ c1] [ c1] F ( ) a b. [ c] 1[ c] or Now c c+1 c c+1 [ c 1] [ c] 1. c1 c c c1 [ ] [ c 1] [ ] [ c 1] b a 1 [ c] 1 1 [ c] [ ] [ ] b 1 1 a 1 1. Thus the proof of Theorem 3. s establshed. 89

10 Saurabh Porwal & Anta Gupta References Aral A., Agarwal R. & Gupta V Applcatons of -calculus n Operator Theory. New Yor, NY: Sprnger. Clune J. & Shel-Small T armonc unvalent functons. Ann. Acad. Sc. Fen. Seres AI Math. 9: 3-5. Dxt K.K. & Chandra V On subclass of unvalent functons wth postve coeffcents. Algarh Bull. Math. 7(): Dxt K.K. & Patha A.L A new class of analytc functons wth postve coeffcents. Indan J. Pure Appl. Math. 34(): Dxt K.K. & Porwal S A subclass of harmonc unvalent functons wth postve coeffcents. Tamang J. Math. 41(3): Dxt K.K., Porwal S. & Dxt A A new subclass of unvalent functons wth postve coeffcents, Bessel J. Math. 3(): Duren P armonc Mappngs n the Plane. Cambrdge: Cambrdge Unversty Press. Patha A. L., Porwal S., Agarwal R. & Msra R. 01. A subclass of harmonc unvalent functons wth postve coeffcents assocated wth fractonal calculus operator. J. Nonlnear Anal. Appl. Artcle ID jnaa-00108, 11 Pages. Porwal S. 015a. On a new subclass of harmonc unvalent functons defned by multpler transformaton. Math. Moravca 19(): Porwal S. 015b. A new subclass of harmonc unvalent functons assocated wth fractonal calculus operator, Fract. Dff. Calc. 5(1): Porwal S. & Aouf M.K On a new subclass of harmonc unvalent functons defned by fractonal calculus operator. J. Frac. Calc. Appl. 4(10): 1-1. Porwal S. & Dxt K.K An applcaton of certan convoluton operator nvolvng hypergeometrc functons. J. Raj. Acad. Phy. Sc. 9(): Porwal S. & Dxt K.K. 01. On a new subclass of Salagean-type harmonc unvalent functons. Indan J. Math. 54(): Porwal S. & Dxt K.K New subclasses of harmonc starle and convex functons. Kyungpoo Math. J. 53: Porwal S., Dxt K.K., Kumar V. & Dxt P On a subclass of analytc functons defned by convoluton. Gen. Math. 19(3): Porwal S., Dxt K.K., Patha A.L. & Agarwal R. 01. A subclass of harmonc unvalent functons wth postve coeffcents defned by generaled Salagean Operator. J. Raj. Acad. Phy. Sc. 11(): Uralegadd B.A., Gang M.D. & Sarang S.M Unvalent functons wth postve coeffcents, Tamang J. Math. 5(3): Uralegadd B.A., Gang M.D. & Sarang S.M Close-to-convex functons wth postve coeffcents. Studa Unv. Babes-Bolay Math. XL(4): Department of Mathematcs Sr Radhey Lal Arya Inter College, Ahan, athras (U.P.), INDIA E-mal: saurabhjcb@redffmal.com* Department of Mathematcs UIET, C.S.J.M. Unversty, Kanpur (U.P.), INDIA-0804 E-mal: anta.an@gmal.com *Correspondng author 90