A New Class of Analytic p-valent Functions with Negative Coefficients and Fractional Calculus Operators

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1 Tamsui Oxford Joural of Mathematical Scieces 20(2) (2004) Aletheia Uiversity A New Class of Aalytic -Valet Fuctios with Negative Coefficiets ad Fractioal Calculus Oerators S. P. Goyal Deartmet of Mathematics, Uiversity of Rajastha, Jaiur , Idia ad J. K. Prajaat Deartmet of Mathematics, Sobhasaria Egieerig College, Goulura, NH-11, Siar , Idia Received August 20, 2003, Acceted December 31, Abstract I this aer we defie a ew class of fuctios which are aalytic ad -valet with egative coefficiets, by usig fractioal differ-itegral oerators studied recetly by authors. Characterizatio, distortio theorems ad other iterestig roerties of this class of fuctios are studied. Some secial cases of mai results are also oited out. Keywords ad Phrases: Aalytic fuctios, -valet fuctios, Fractioal differitegral oerators, Distortio roerty.

2 176 S. P. Goyal ad J. K. Prajaat 1. Itroductio Let T (, ) deote the class of fuctios f (z) of the form f (z) = z a z ( a 0;, N = { 1,2,3, L } ) (1.1) = + which are aalytic ad multivalet (-valet) i the oe uit dis U = { z : z C ad z < 1 } (1.2) μ A fuctio of f (z) T (, ) is said to be i the class J, νη, ( abσ ) if ad oly if S 0, z f (z) 1 b S f (z) a < σ ( z U) (1.3) for 1 a < b 1 ; 0 < b 1 ; 0 < σ 1, < μ < 1, ν < + 1, η R +, N μ ad S, νη, is the fractioal differitegral oerator of order μ (- < μ < 1) (see Goyal ad Prajaat [1] ). For this oerator If the S : T (, ) T (, ) (1.4) φ μ ν η = + S f (z) = z (, ) a z (1.5) ( a 0;, N = { 1,2,3, L }, z U ) where φ (, ) = L (, ) M (,) (1.6) Here ad L (, ) M (,) (1 ν ) ( 1 μ+ η) = ( 1 ν + η) Γ ( + 1) Γ ( + 1) ( 1 ν + η ) = ( 1 ν ) ( 1 μ+ η) (1.7) (1.8) Throughout the aer ( a) = ( a + 1), or = a (a + 1) (a + 2) L (a + + 1) (1.9) = 1 is the Factorial fuctio, or if a > 0, the Γ (a + ) (a) = Γ ( a) ( where Γ is Euler's Gamma fuctio). (1.10)

3 A New Class of Aalytic -Valet Fuctios 177 For z 0, (1.5) may be exressed as S f (z) Γ(1 ν + ) Γ(1 μ + η + ) z ν J μ, ν, η f (z) ; 0 μ < 1 Γ (1 + ) Γ(1 ν + η + ) = Γ (1 ν + ) Γ (1 μ + η + ) z ν I μ, ν, η f (z) ; < μ < 0 Γ (1 + ) Γ(1 ν + η + ) (1.11) μ where J, νη, f (z) is the fractioal derivative oerator of order μ (0 μ < 1), while μ, νη, I f (z) is the fractioal itegral oerator of order - μ (- < μ < 0) itroduced ad studied by Saigo ( [4], [5] ). becomes It may be worthotig that, by choosig - < μ = ν < 1 the oerator μμη μ Γ (1 μ + ) μ μ z z S μ νη f (z) S f (z) = S f(z) = z D f (z) (1.12) Γ ( + 1) where D μ z f (z) is resectively, the fractioal itegral oerator of order - μ (- < μ < 0) ad fractioal derivative oerator of order μ (0 μ < 1) cosidered by Owa [3] ad defied by Liouville [2]. Further if μ = ν = 0, the S f (z) = f (z) (1.13) 0,0, η 0, z ad for μ 1 ad ν = 1 ' lim S f (z) = z f ( z) (1.14) μ 1 μ,1, η Note that (i) For μ 1 ad ν = 1 ad σ = 1, we have lim J (a,b,1) = τ (, a, b) (1.15) μ 1 μ,1, η where τ (, a, b) is the subclass of fuctios f (z) T (, ) satisfyig the coditio ' zf ( z) 1 < 1 (1.16) ' bz f ( z) a ( 1 a < b 1, 0 < b 1, N, z U ) The class τ (, a, b) was studied by Shula ad Dashrath [6] i a slight differet form.

4 178 S. P. Goyal ad J. K. Prajaat (ii) For < μ = ν < = α = σ reduces to 1, a 2 1, b 1, J μνη ( a, b, ) J ( 2α 1,1, σ) = T ( α, σ, μ) (1.17) μμη where T ( α, σ, μ ) is the subclass of fuctios satisfyig the coditios μ Sz f (z) 1 μ S f (z) + (1-2 α ) z < σ ( z U, 0 α < 1, 0 < σ 1, < μ < 1, N ad η R+ ) (1.18) Here S μ z is give by (1.12), the class of fuctios T (α, σ, μ) was studied by Srivastava ad Aouf ( [7] ad [8] ) uder restricted coditio for μ (0 μ < 1). Our urose i this aer is to obtai certai roerties for J μ νη ( abσ ) such as coefficiet bouds, iclusio theorems, closure roerty ad distortio roerties. 2. Coefficiet Characterizatio Theorem Theorem 1. Let fuctio f (z) defied by (1.1) be i class T (, ) the the fuctio f(z) belogs to the class J μ νη 0, z ( abσ ) ( -1 a < b 1,0 < b 1, 0 < σ 1, - < μ < 1, ν < + 1, η R +, N ) iff φ ( μ, ν, η, ) (b σ + 1) a σ ( b a) (2.1) = + Proof. Let the iequality (2.1) holds true, the S f (z) 1 σ b S f (z) a 0, z = z φ ( μ, ν, η, )a z 1 σ bz bφ ( μ, ν, η, )a z a = + = + φ ( μ, ν, η, )(bσ + 1) a σ ( b a) = + 0 [ by (2.1) ] Hece by maximum module theorem f (z) belogs to class J μ νη ( abσ ).

5 A New Class of Aalytic -Valet Fuctios 179 Now to rove coverse, we assume that f(z) is defied by (1.1) ad f (z) J ( a, b, σ ). The by usig (1.5) i (1.3), we get S z )a z μνη φ μ ν η 0, z f (z) 1 = + = 0, z φ ( μ, ν, η, = + (, 1 b S f (z) a b z )a z a Sice Re (z) z, therefore we have Re z φ ( μ, ν, η, )a z 1 = + σ < b z b φ ( μ, ν, η, )a z a = + (2.2) Now lettig z 1, through real values i (2.2), we at oce obtai (2.1) ad theorem is comletely roved. Corollary 1. Let a = 2α - 1, b = 1 ad - < μ = ν < 1, i theorem 1. The a fuctio f (z), as i (1.1), is i class T (α, σ, μ) (- < μ < 1, 0 α < 1, 0 < σ 1, N, η R + ) iff Γ ( + 1) (1 μ) ( σ + 1) a 2 σ (1 α) (2.3) Γ ( + 1) (1 μ) = + The result (2.3), for the class of fuctios T (α,σ, μ) was studied by Srivastava ad Aouf ( [7] ad [8] ), uder the coditio for μ (0 μ < 1). Remar. Similarly o taig μ 1, ν = 1 ad σ = 1 i theorem 1 ad by usig (1.14), the coefficiet characterizatio theorem for the class τ (, a, b) is obtaied ad thereby results due to Shula ad Dashrath [7] become articular cases of our theorem.

6 180 S. P. Goyal ad J. K. Prajaat 3. Growth ad Distortio Theorems μνη Theorem 2. Let the fuctio f(z) of (1.1) be i class J ( a, b, σ ) uder the coditios of validity 1 a < b 1,0 < b 1, 0 < σ 1, < μ < 1, ν < + 1, ν η η R+, N ad < μ < 1 ad if further, for β < + 1, γ R+, α > ν ad satisfyig the iequality the β ( α + γ) ( + 1) (3.1) α α, β, γ σ ( b α) (1 β + γ + ) (1 ν + ) (1 μ+ η+ ) S f (z) z z (bσ + 1) (1 β + ) (1 + α + γ + ) (1 ν + η+ ) ad + α, β, γ σ ( b a) (1 β + γ + ) (1 ν + ) (1 μ+ η+ ) S f (z) z + z (bσ + 1) (1 β + ) (1 + α + γ + ) (1 ν + η+ ) + Equatios (3.2) ad (3.3) are justified for σ ( b a)(1 ν + ) (1 μ+ η+ ) f (z) = z z ( bσ + 1)( + 1) (1 ν + η+ ) + (3.2) (3.3) (3.4) Proof. Uder the assumtio ad coditios of validity for the theorem it is obvious that the fuctio φ ( μ, ν, η, ) = is icreasig for +. ( 1 ν ) (1 μ+ η) Γ ( + 1) (1 ν + η) (1 ν + η) Γ ( + 1)(1 ν) (1 μ+ η) Ideed ( + 1) (1 ν + η+ ) φ+ 1( μ, ν, η, ) φ( μ, ν, η, ) = φ( μ, ν, η, ) 1 (1 ν + ) (1 μ+ η+ ) μ( + 1) ν ( μ η) = φ ( μ, ν, η, ) > 0 (1 ν + ) (1 μ+ η+ )

7 A New Class of Aalytic -Valet Fuctios 181 or φ (, ) φ (, ) μ( + + 1) ν ( μ η) = φ+ ( μ, ν, η, ) > 0 (1 ν + + ) (1 μ+ η+ + ) Thus φ ( μ, ν, η, )(bσ + 1) a φ ( μ, ν, η, )(bσ + 1) σ ( b a) + = + = + or equivaletly σ ( b a) (1 ν + ) (1 μ+ η+ ) a (3.5) ( bσ + 1)( + 1) (1 ν + η+ ) = + S α βγ 0, z Now usig (1.5), the oerator for f (z) may be exressed as α, β, γ φ α β γ = + S f (z) = z (, )a z ( a 0) (3.6) The assumtio (3.1) comrehesively sells that (1 β ) (1 + α + γ) Γ ( + 1)(1 β + γ) φ ( α, β, γ, ) = (1 β + γ) Γ ( + 1) (1 β) (1 + α + γ) is o icreasig for +, ad thus, we have ( + 1) (1 β + γ + ) 0 φ ( αβγ, ) φ+ ( αβγ, ) = (1 β + ) (1+ α + γ + ) Now emloyig (3.5) ad (3.8) i (3.6) we ote that α, β, γ + 0, z φ+ ( α, β, γ, = + S f (z) z z ) a (1 β + γ + ) (1 ν + ) (1 μ+ η+ ) σ ( b a) z z (1 β + ) (1 + α + γ + ) (1 ν + η+ ) ( bσ + 1) + (3.7) (3.8) which is icidetally asserts (3.2) of theorem 2. Assertio (3.3) follows a similar aalysis. Corollary 2. Let we set β = - α, a = 2α - 1, b = 1 ad - < μ = ν < 1 i theorem 2. Now if f (z) be as i class T (α, σ, μ) (- < μ < 1, 0 α < 1, 0 < σ 1, N, η R + ) the α 2 σ (1 α)(1 μ+ ) + Sz f (z) z z (3.9) ( σ + 1)(1+ α + )

8 182 S. P. Goyal ad J. K. Prajaat ad α 2 σ (1 α)(1 μ+ ) Sz f (z) z + z ( σ + 1)(1+ α + ) + where α > 0 ad iequality (3.9) ad (3.10) are justified if 2 σ (1 α)(1 μ+ ) f (z) = z z ( σ + 1)(1+ ) + (3.10) (3.11) Remar. O taig μ 1, ν = 1ad σ = 1i Theorem 2, ad usig (1.14), the distortio theorem for the class τ (, a, b) is obtaied, ad thereby result due to Shula ad Dashrath [6] i slight differet form, become articular case of our theorem. 4. Closure Theorem Let the fuctio f i (z) be defied for i = 1,L, m by fi( z) = z ai, z ( ai, 0, N ad z U ) (4.1) = + Now we rove a result for the closure uder Arithmatic mea of fuctio i the class J ( a, b, σ ). Theorem 3. Let the fuctio f i (z) defied by (4.1) be i class each i = 1,, m the the fuctio K (z) defied by is i class 1 K (z) = z ai, z m = + i = 1 J ( a, b, σ ), where J ( a, b, σ ) for i i (4.2) a = mi { a }, b = max{ b } (4.3) i 1 i m 1 i m i μνη Proof. Let f i (z) J ( a, b, σ ) for each i = 1,L, m the by (2.1) we have i i m m 1 1 ( bi ai) σ ( b a) σ φ ( μ, ν, η, ) ai, m m bσ + 1 b σ + 1 = + i = 1 i = 1 i where a ad b are defied by (4.3). This imlies that K (z) J ( a, b, σ ).

9 A New Class of Aalytic -Valet Fuctios 183 μ,ν,η 5. Further Proerties of ( a,b, σ) J Theorem 4. If < ξ ν < + 1, < λ μ < 1 ad η R+, 1 a < b 1, 0 < b 1, 0 < σ 1ad N the J ( a, b, σ ) J ( a, b, σ ) (5.1) λξη Proof. Let f (z) J ( a, b, σ ) the by Theorem 1 (1 ν ) (1 μ+ η) Γ ( + 1) (1+ ν + η) ( bσ + 1) a < σ ( b a) (5.2) (1 ν + η) Γ ( + 1) (1 ν) (1 μ+ η) = + If we write (1 ) (1 ) (1 ) ( ) ν ν + η ad ( ) μ + φν = ϕ μ = η (1 ν + η) (1 ν) (1 μ+ η) we observe that (2 ν ) (2 ν + η) (1 ν) (1 ν + η) φν ( 1) φν ( ) = (2 ν + η) (2 ν) (1 ν + η) (1 ν) (5.3) Γ(1 ν + ) Γ(1 ν + η+ ) (1 ν + ) (1 ν + η+ ) = 1 Γ(1 ν + η+ ) Γ(1 ν + ) (1 ν + η+ ) (1 ν + ) Γ(1 ν + ) Γ(1 ν + η+ ) η ( ) = < 0 (5.4) Γ(1 ν + η+ ) Γ(1 ν + ) (1 ν + η+ ) (1 ν + ) which states that φ (ν) beig a icreasig fuctio of ν (- <ν < + 1) ad similarly, we ca show that ϕ (μ) is icreasig fuctio of μ, such that - < μ < 1 therefore φ ( λξη, )(bσ + 1) a φ (, ) (bσ + 1) a < ( b a) σ (5.5) = + = + which imlies that f J ( a, b, σ ) ad thereby theorem is established. λξ η

10 184 S. P. Goyal ad J. K. Prajaat Theorem 5. If 1 a < b 1, < μ < 1, ν < + 1, η R+, 0 < b 1, 0 < σ 1, N ad 1 a' < b' 1, 0 < b' 1, the J ( a, b, σ ) = J ( a', b', σ ) (5.6) if ad oly if σ ( ab' ba') = (a' a) ( b' b) (5.7) Proof. From (2.1) of theorem, we have b σ + 1 b' σ + 1 φ ( μ, ν, η, ) a = φ ( μ, ν, η, ) a 1 σ ( b a) σ ( b' a') = + = + which imlies that b σ + 1 b' σ + 1 = σ ( b a) σ ( b' a') (5.8) immediately yields (5.7). (5.8) Remar. O uttig b' = 1 i Theorem 5, we get uder the coditios metioed therei 1+ a b a σ J ( a, b, σ ) = J,1, σ (5.9) b σ + 1 Theorem 6. If 1 a 1 a 2 < b 1, < μ < 1, ν < + 1, η R+, 0 < b 1, 0 < σ 1, ad N the J ( a, b, σ ) J ( a, b, σ ) (5.10) 1 2 Proof. Let f J ( a, b, σ ), them from (2.1) of theorem 1, we have = + φ ( μ, ν, η, )a σ ( b a2) σ ( b a1) ( b σ + 1) ( b σ + 1) uder the coditio 1 a a 1 which imlies that 1 2 f (z) J ( a, b, σ ) ad we get (5.10).

11 A New Class of Aalytic -Valet Fuctios 185 Theorem 7. If 1 a b 1 b 2 1, < μ < 1, ν < + 1, η R+ ad N the J ( a, b, σ ) J ( a, b, σ ) (5.11) 1 2 Proof. Theorem follows by similar aalysis as i theorem 6. Theorem 8. 1 a < a < b b 1, < μ < 1, ν < + 1, 0 < b b 1, η + R ad N the J ( a, b, σ ) J ( a, b, σ ) (5.12) Proof. Let f (z) J ( a, b, σ ) the by (2.1) of Theorem 1, we have 2 1 φ ( μ, ν, η, )a ( b1 a2) σ ( b 2 a1) σ ( b σ + 1) ( b σ + 1) = Now uder the coditios stated i the theorem, we get which gives (5.12). f (z) J ( a, b, σ ), 2 1 Acowledgemet The first author (S.P.G.) is thaful to the Uiversity Grats Commissio, New Delhi, for rovidig some fiacial assistace. The authors are also thaful to the worthy referee for his useful suggestios for imrovemet of the aer.

12 186 S. P. Goyal ad J. K. Prajaat Refereces [1] Goyal,S.P. ad Prajaat, J.K., New class of aalytic fuctios ivolvig certai fractioal differ-itegral oerators, Proceedigs of 4 th Aual Coferece of the Society for Secial Fuctios ad their Alicatios. (Eds: A.K.Agarwal, M.A.Patha ad S.P.Goyal), 4 (2004), [2] Liouville,J., Memoire sur le Calcul des differetielles a idices quelcoques, J. Ecole Polytech. 13 (1832), [3] Owa,S. O the distortio theorems I, Kyugoo Math. J. 18 (1978), [4] Saigo, M., A remar o itegral oerators ivolvig the Gauss hyergeometric fuctios, Math. Re. College Geeral Ed. Kyushu Uiv. 11 (1978), [5] Saigo,M., A certai boudary value roblem for the Euler-Darboux equatio, Math. Jao, 25 (1979), [6] Shula,S.L. ad Dashrath, O the certai classes of multivalet fuctios with egative coefficiets, Idia J. Pure Al. Math. 20 (1984), [7] Srivastava,H.M. ad Aouf, M.K., A certai fractioal derivative oerator ad its alicatios to a ew class of aalytic ad multivalet fuctios with egative coefficiets. I, J. Math. Aal. Al. 171 (1992), [8] Srivastava,H.M. ad Aouf, M.K., A certai fractioal derivative oerator ad its alicatios to a ew class of aalytic ad multivalet fuctios with egative coefficiets. II, J. Math. Aal. Al. 192 (1995),