SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi

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1 Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: Float 3:3 (009), DOI:098/FIL J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE Sayal S Josh ABSTRACT I the preset paper a ew subclass of aroc uvalet fuctos s defed usg Salegao dervatve operator ad several terestg propertes le coeffcet boud, dstorto theore are obtaed 000 Matheatcs Subject Classfcato:30C45,30C55 Keywords ad phrases:aroc fuctos,uvalet,sece-preservg,aalytc Receved: Jue 5, 009 Coucated by Draga S Djordjevć Itroducto A cotuous fucto f s sad to be coected doa coplex plae D Such fuctos ca be expressed as a coplex-valued haroc fucto a sply I f are real haroc D f both Re{ f } ad { } f = h+ g () h ad g D g f A ecessary ad suffcet codto for f to be locally uvalet ad D s that h ( z) > g ( z) for all z D, see[ ] where are aalytc We call h as aalytc part ad as coaalytc part of sese-preservg Let S be the faly of fuctos of the for () that are haroc,uvalet ad oretato preservg the ope ut ds U = { z: z < }, so that f = h+ g s oralzed by f(0) = h(0) = f z (0) = 0 Further f = h+ g ca be uquely detered by the coeffcets of power seres expasos = = hz = z+ az, gz = bz, z U, b <, () a =,3,4, where for ad b for =,,3, 303

2 We ote that ths faly S was vestgated ad studed by Clue ad Shel-Sall [ ] ad t reduces to the well-ow faly S, the class of all oralzed aalytc uvalet fuctos gve (), wheever the co-aalytc part of Let where h g f s detcally zero S deote the subfaly of S cosstg of haroc fuctos of the for f = h+ g = = hz = z+ az, g ( z) = bz, z U, b <, (3) Recetly, Jahagr et al [5] defed the Salagea dervatve of haroc fuctos f h g S by = + D f( z) = D h( z) + ( ) D g( z) (4) N {0}, where Salagea dervatve of power seres φ( z) = φz D 0 φ( z) φ( z) =, D ( z) z ( z) ad φ = φ = s gve by ( D φ z D D φ( z)) φ z = = = Defto The fucto f = h+ g defed by () s the class S (, ; ) + + D f( z) D f( z) + f Re (5) D f( z) D f( z) where 0 <,0 <, N {0} Also let S (, ; ) S (, ; ) S (6) = We ote that by specalzg the paraeter, especally whe 0 =, S (, ; ) reduces to well-ow faly of starle haroc fuctos of order I recet years ay researchers have studed varous subclasses of S for exaple [],[3],[4],[7]ad [8] I the preset paper we a at systeatc study of basc propertes, partcular coeffcet boud, dstorto theore ad extree pots of aforeetoed subclass of haroc fuctos Ma Results Theore Let f = h+ g be gve by () If codto + a b = where a =, 0 <, 0 <, N {0} () the f s sese-preservg haroc uvalet U ad f S (, ; ) Proof If the equalty () holds for coeffcets of f = h+ g the by (), f s oretato preservg ad haroc uvalet U Now t reas to show that 304

3 f S (, ; ) Accordg to (4) ad (5) we have + + D f( z) D f( z) Re + D f( z) D f( z) Re Az Bz whch s equvalet to > where + A( z) ( ) D f z D f z ad B z D f = + = z Usg the fact that, Re( w) Az + Bz Az ( + ) Bz substtutg values of > f + w + w t suffces to show that Az ad B( z ) ad wth sple calculatos we led to [ ] [ ] ( ) z+ ( + ) + ( ) a z ( ) ( + ) ( ) b z = = z [( ) ] a z [( ) ] b z = = [ ] [ ] z ( + ) a z ( ) ( + ) + + b z ( ) z 0 = = ( + ) ( + ) + + a z b z = = by assupto ece proof s copleted The fuctos f ( z) = z+ x z + y z ( + ) ( + ) + + [ ] [ ] = = where x + y =, (3) = = shows that the coeffcet boud gve () s sharp Theore Let f = h+ g be so that h ad g are gve by (6) The f S (,, ) f ad oly f [( + ) ] [( + ) + + ] a + b = where a =, 0 <, 0 <, N {0} (4) Proof The f part follows for Theore wth the fact that S (,, ) S (,, ) 305

4 For oly f part, we show that f S (,, ) f the codto (4) s ot satsfed Note that ecessary ad suffcet codto for f = h+ g gve by (6) to be S (,, ) s that + + D f( z) D f( z) Re + D f( z) D f( z) whch s equvalet to Re =Re >0 + ( + ) D f( z) + ( ) D f( z) D f ( z) ( ) [( + ) ] ( ) [( + ) + + ] z az bz = = z az + bz = = The above codtos ust hold for all values of z, z = r < Choosg z o postve axs where 0 z = r < we have ( ) [( + ) ] ( ) [( + ) + + ] z ar br = = z ar + br = = (5) or equvaletly f the codto (4) dose ot hold the the uerator (5) s egatve for r suffcetly close to Thus there exsts z = r (0,) for whch the quotet (5) s egatve Ths cotradcts that requred codto for Theore3 Let 0 0 f S (,, ) ad hece proof s copleted f be gve by (6) The f S (,, ) f ad oly f = ( + ) f z x h z y g z = 0 306

5 where h ( z) = z, h ( z) = z z, =,3, [( + ) ] g z = z+ z = [( + ) + + ] ad,,,3, x 0, y 0, x = ( x + y ) 0 = I partcular, the extree pots of S (,, ) Proof Let = ( + ) f z x h z y g z = are { } g h ad{ } = x + y z x z + y z [ ] = = + [( + ) + + ] The ( + ) ( + ) + + a + [ ] [ ] = = = x + y = x = = ad so f S (,, ) Coversely, suppose that f S (,, ) Settg x y [( + ) ] = a, =,3, [( + ) + + ] = b, =,, where ( x + y) =, we obta f( z) = ( xh( z) + yg ( z )) = = as requred Theore 4 Let f S (,, ) the for z = r < we have ( + ) + + f ( z) ( + b ) r + b ( + ) ( + ) r ad b 307

6 + + + f ( z) ( b ) r b ( + ) + Proof Let f S(,, ) Tag absolute value of f we obta f ( z) ( + b ) r + ( a + b ) r = ( + b ) r + ( a + b ) r = r ( + ) ( + b ) r + ( a b ) r + ( ) + = ( + ) ( ) ( + b ) r + a (( ) + + ) = ( ) ( + b ) r + b ) r (( ) + ) = ( + b ) r + b r (( + ) ) + b ) r The forthcog result follows fro left had equalty Theore 4 Corollary Let the f of the for (3) be so that f f S (,, ), ω; ω < b f ( U) ( + ) ( + ) Theore 5 The class S (,, ) s closed uder covex cobato Proof For =,, suppose f ( z) S (,, ) where = + = = f z z a z b z the by Theore ( ) = = ( ) a + b (6) 308

7 For t = =, 0 t, the covex cobato of f ay be wrtte as tf z z t a z t b z = + = = = = = hece by (6) (( + ) ) ( ) t a t b = + = = = (( + ) ) (( + ) + + ) t a b = + = = = = ad therefore tf ( z) S (,, ) = Ths copletes the proof Acowledgeet Ths wor s supported by Departet of Scece ad Techology, SERC Dvso, New Delh uder Youg Scetst Project (SR/FTP/MS-7/007) sactoed to author Refereces OPAhuja, R Aghalary ad S B Josh, aroc uvalet fuctos assocated wth - uforly starle fuctos, Math Sc ResJ,9 () (005), 9-7 J Clue, T Shel-Sall, aroc uvalet fuctos, AAcad Sc Fe AIMath, 9 (984), J M Jahagr, aroc fuctos starle the ut dsc, JMath Aal Appl, 35 (999), J M Jahagr, Y C K ad M Srvastava, Costructo of a certa class of haroc close-to covex fuctos assocated wth Alexader tegral trasfor, Itegral Tras ad Spec Fuct, 4 (003), J M Jahagr, G Murgusudaroorthy ad K Vjay, Salagea type haroc uvalet fuctos, South JPure ad Appl Math, (00), G S Salagea, Subclass of uvalet fuctos, Lect Notes Math Sprger-Verlag, 03 (983), Slvera, aroc uvalet fuctos wth egatve coeffcets, J Math Aal Appl,0 (998), S Yalc, O certa haroc uvalet fuctos defed by Salagea dervatve, Soochow JMath, 3 (3) (005), 3-33 t Departet of Matheatcs, Walchad College of Egeerg, Sagl 46 45, Maharshtra, Ida E-al: sayal_75@yahooco 309