Tamsui Oxford Joural of Mathematical Scieces 23(2 (27 141-153 Aletheia Uiversity O Certai Subclass of λ-bailevič Fuctios of Type α + iµ Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua College of Mathematics ad Computig Sciece Chagsha Uiversity of Sciece ad Techology Chagsha, Hua, 4176, People s epublic of Chia eceived March 26, 25, Accepted October 5, 25. Abstract I the preset paper, the authors itroduce a ew subclass B (λ, α, µ, A, B, of λ Bailevič fuctios of type α + iµ. The subordiatio relatios ad iequality properties are discussed by makig use of differetial subordiatio method. The results preseted here geeralie ad improve some kow results, ad some other ew results are obtaied. Keywords ad Phrases: λ-bailevič fuctios of type α + iµ, Differetial subordiatio. 1. Itroductio ad Defiitios Let A deote the class of fuctios of the form f( = + k=+1 2 Mathematics Subject Classificatio. Primary 3C45. E-mail: higwag @163.com E-mail: cygao1 @163.com E-mail: shaomouyua @163.com a k k ( N = {1, 2, 3,..., (1.1
142 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua which are aalytic i the uit disk U = { : < 1. Also let S (β deote the usual class of starlike fuctios of order β, β < 1. Let f( ad F ( be aalytic i U. The we say that the fuctio f( is subordiate to F ( i U, if there exists a aalytic fuctio ω( i U such that ω( ad f( = F (ω(, deoted by f F or f( F (. If F ( is uivalet i U, the the subordiatio is equivalet to f( = F ( ad f(u F (U (see [1]. The followig class of aalytic fuctios were studied by various authors (see [3]. Defiitio 1. Let B (α, µ, β, g( deote the class of fuctios i A satisfyig the iequality f ( α+iµ ( f( > β ( U, (1.2 f( g( where α, µ, β < 1 ad g( S (β. The fuctio f( i this class is said to be Bailevič fuctio of type α + iµ ad of order β. I the preset paper, we defie the followig class of aalytic fuctios. Defiitio 2. Let B (λ, α, µ, A, B, g( deote the class of fuctios i A satisfyig the iequality (1 λ f ( f( ( α+iµ ( f( + λ 1 + f ( g( f ( ( f α+iµ ( 1 + A g ( 1 + B ( U, (1.3 where λ 1, α, µ, 1 B 1, A B, A ad g( S (β. All the powers i (1.3 are pricipal values, below we apply this agreemet. The fuctio f( i this class is said to be λ Bailevič fuctio of type α+iµ. If α = 1, µ =, A = 1 2β ad B = 1, the the class B (λ, α, µ, A, B, g( reduces to the class of λ close-to-covex fuctios of order β, β < 1. If α =, µ =, A = 1 ad B = 1, the the class B (λ, α, µ, A, B, g( reduces to the class of λ covex fuctios [6]. If α =, µ =, A = 1 2β ad B = 1, the the class B (λ, α, µ, A, B, g( reduces to the class of λ covex fuctios of order β, β < 1. If A = 1 2β ad B = 1, the the class B (λ, α, µ, A, B, g( reduces to the class of λ Bailevič fuctios of type α + iµ ad of order β, β < 1. Li [3], Owa [4], Owa ad Nuokawa [5] discussed the related properties of
O Certai Subclass 143 the classes B (λ, α, µ, 1 2β, 1,, B (, α,, 1 2β, 1, ad B (λ, 1,, 1 2β, 1,, respectively. I the preset paper, we will discuss the subordiatio relatios ad iequality properties of the class B (λ, α, µ, A, B,. The results preseted here geeralie ad improve some kow results, ad some other ew results are obtaied. 2. Prelimiaries esults I order to establish our mai results, we shall require the followig lemmas. Lemma 1 ([7]. Let F ( = 1 + b + b +1 +1 + be aalytic i U, h( be aalytic ad covex i U, h( = 1. If where c ad c, the F ( + 1 c F ( h(, (2.1 F ( c c t c 1 h(tdt h(, ad c c t c 1 h(tdt is the best domiat for (2.1. Lemma 2 ([8]. Let f( = k=1 a k k be aalytic i U, g( = k=1 b k k be aalytic ad covex i U. If f( g(, the a k b 1, for k = 1, 2,.... Lemma 3. Let λ 1, α, µ, α + iµ, 1 B 1, A B ad A. The f( B (λ, α, µ, A, B, if ad oly if q( + 1 α + iµ q ( 1 + A 1 + B, (2.2 where q( = (1 λ (f(/ α+iµ + λ(f ( α+iµ. Proof. Let ( α+iµ f( = m(. (2.3
144 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua The, by takig the derivatives i the both sides of (2.3, we have that is, f ( f( f ( f( ( f( ( α+iµ ( f( f( = α+iµ = m( + α + iµ m (, α+iµ + α + iµ Substitutig f( by f ( i (2.4, we have ( 1 + f ( f ( ( α+iµ f(. (2.4 (f ( α+iµ = (f ( α+iµ + ( (f ( α+iµ. (2.5 α + iµ From equalities (2.4 ad (2.5, we get = (1 λ f ( α+iµ ( ( f( + λ 1 + f ( (f ( α+iµ f( f ( ( α+iµ f( (1 λ + λ (f ( α+iµ + ( α+iµ f( (1 λ + λ (f ( α+iµ. (2.6 α + iµ Now, suppose that f( B (λ, α, µ, A, B,, ad let q( = (1 λ ( α+iµ f( + λ (f ( α+iµ. Thus, from the defiitio of B (λ, α, µ, A, B, ad equality (2.6, we ca get (2.2. O the other had, this deductive process ca be coverse. Therefore, the proof of Lemma 3 is complete.
O Certai Subclass 145 3. Mai esults ad Their Proofs Theorem 1. Let λ 1, α, µ, α + iµ, 1 B 1, A B ad A. If f( B (λ, α, µ, A, B,, the (1 λ ( f( α+iµ + λ (f ( α+iµ α + iµ 1 + Au 1 + Bu u α+iµ 1 du 1 + A 1 + B. Proof. First let q( = (1 λ (f(/ α+iµ + λ (f ( α+iµ, the q( = 1 + b +b +1 +1 + is aalytic i U. Now, suppose that f( B (λ, α, µ, A, B,, by Lemma 3, we kow q( + 1 α + iµ q ( 1 + A 1 + B. It is obvious that h( = (1 + A/(1 + B is aalytic ad covex i U, h( = 1. Sice α + iµ ad α, therefore it follows from Lemma 1 that (1 λ ( α+iµ f( + λ (f ( α+iµ α + iµ α+iµ t α+iµ 1 h(tdt = α + iµ 1 + Au 1 + Bu u α+iµ 1 du 1 + A 1 + B. Corollary 1. Let λ 1, α, µ, α + iµ ad β 1. If f( A satisfies (1 λ f ( f( the (1 λ ( f( ( α+iµ ( f( +λ 1 + f ( (f ( α+iµ f ( ad (3.1 is equivalet to (1 λ ( f( α+iµ +λ (f ( α+iµ α + iµ 1 + (1 2βu 1 + (1 2β 1 ( U, u α+iµ 1 du ( U, (3.1 α+iµ +λ (f ( α+iµ (1 β(α + iµ 1 β+ u α+iµ 1 du ( U.
146 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua Theorem 2. Let λ 1, α, µ, α + iµ, 1 B 1, A B ad A. If f( B (λ, α, µ, A, B,, the { α + iµ ( if 1 + Au U 1 + Bu u α+iµ α+iµ f( 1 du < (1 λ + λ (f ( α+iµ { α + iµ 1 + Au < sup U 1 + Bu u α+iµ 1 du. Proof. Suppose that f( B (λ, α, µ, A, B,, from Theorem 1 we kow ( α+iµ f( (1 λ + λ (f ( α+iµ α + iµ 1 + Au 1 + Bu u α+iµ 1 du. Therefore it follows from the defiitio of the subordiatio that ( α+iµ f( { (1 λ + λ (f ( α+iµ α + iµ > if 1 + Au U 1 + Bu u α+iµ 1 du, ad ( α+iµ f( { (1 λ + λ (f ( α+iµ α + iµ < sup U 1 + Au 1 + Bu u α+iµ 1 du. Corollary 2. Let λ 1, α, µ, α + iµ ad β < 1. If f( B (λ, α, µ, 1 2β, 1,, the { α + iµ β + (1 β if U u α+iµ 1 du ( α+iµ f( < (1 λ + λ(f ( α+iµ { α + iµ < β + (1 β sup U u α+iµ 1 du. Corollary 3. Let λ 1, α, µ, α + iµ ad β > 1. If f( A satisfies ( α+iµ ( (1 λf ( f( + λ 1 + f ( (f ( α+iµ < β ( U, f( f (
O Certai Subclass 147 the β + (1 β sup < U (1 λ ( f( { α + iµ < β + (1 β if U { α + iµ u α+iµ α+iµ + λ(f ( α+iµ 1 du u α+iµ 1 du. Theorem 3. Let λ 1, α ad 1 B < A 1. If f( B (λ, α,, A, B,, the { ( α 1 α 1 Au 1 Bu u α f( 1 du < (1 λ + λ (f ( α < α 1 + Au 1 + Bu u α 1 du ( U, (3.2 ad iequality (3.2 is sharp, with the extremal fuctio defied by (1 λ ( f( α + λ (f ( α = α 1 + Au 1 + Bu u α 1 du. (3.3 Proof. Suppose that f( B (λ, α,, A, B,, from Theorem 1 we kow (1 λ ( f( α + λ (f ( α α 1 + Au 1 + Bu u α 1 du. Therefore it follows from the defiitio of the subordiatio ad A > B that { (1 λ ( α f( + λ (f ( α < sup U α < α { α sup U 1 + Au 1 + Bu u α 1 du { 1 + Au u α 1 du 1 + Bu 1 + Au 1 + Bu u α 1 du,
148 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua ad { ( α f( (1 λ + λ (f ( α { α > if 1 + Au U 1 + Bu u α 1 du α { 1 + Au if u α 1 du U 1 + Bu > α 1 Au 1 Bu u α 1 du. It is obvious that iequality (3.2 is sharp, with the extremal fuctio defied by equality (3.3. Corollary 4. Let λ 1, α ad β < 1. If f( B (λ, α,, 1 2β, 1,, the α 1 { 1 (1 2βu u α 1 du < (1 λ < α ad iequality (3.4 is equivalet to β + (1 βα ( α f( + λ (f ( α 1 + (1 2βu u α 1 du ( U, (3.4 u α 1 du < { ( α f( (1 λ + λ (f ( α < (1 βα 1 β + u α 1 du ( U. Corollary 5. Let α ad β < 1. If f( A satisfies {( 1 + f ( f ( (f ( α > β ( U, the α 1 < α 1 (1 2βu u α 1 du < { (f ( α 1 + (1 2βu u α 1 du ( U, (3.5
O Certai Subclass 149 ad iequality (3.5 is equivalet to β + (1 βα < β + (1 βα u α 1 du < { (f ( α u α 1 du ( U, iequality (3.5 is sharp, with the extremal fuctio defied by f α,β ( = ( α 1 1 + (1 2βt u u α 1 t 1 du u α dt. (3.6 Corollary 6. Let λ > ad β < 1. If f( B (λ, 1,, 1 2β, 1,, the for = r < 1, we have {f ( > 1 λ t 1 λ 1 1 (1 2βt dt. 1 + t emark 1. Corollary 6 is the correspodig result obtaied by Owa ad Nuokawa i [5]. By applyig the similar method as i Theorem 3, we have Theorem 4. Let λ 1, α ad 1 A < B 1. If f( B (λ, α,, A, B,, the { ( α 1 α 1 + Au 1 + Bu u α f( 1 du < (1 λ + λ (f ( α < α 1 Au 1 Bu u α 1 du ( U, (3.7 ad iequality (3.7 is sharp, with the extremal fuctio defied by equality (3.3. Corollary 7. Let λ 1, α ad β > 1. If f( A satisfies { (1 λ f ( f( ( α ( f( + λ 1 + f ( (f ( α < β ( U, f (
15 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua the α 1 { 1 + (1 2βu u α 1 du < (1 λ < α ad iequality (3.8 is equivalet to β + (1 βα ( α f( + λ (f ( α 1 (1 2βu u α 1 du ( U, (3.8 u α 1 du < { ( α f( (1 λ + λ (f ( α < (1 βα 1 β + u α 1 du ( U. Corollary 8. Let α ad β > 1. If f( A satisfies the {( α 1 < α 1 + f ( f ( (f ( α < β ( U, 1 + (1 2βu u α 1 du < { (f ( α ad iequality (3.9 is equivalet to β + (1 βα < β + 1 (1 2βu u α 1 du ( U, (3.9 (1 βα u α 1 du < { (f ( α u α 1 du ( U, iequality (3.9 is sharp, with the extremal fuctio defied by equality (3.6. emark 2. Corollary 7-8 improve the correspodig results of Corollary 6-7 i [3], respectively. If ω, the (ω 1 2 ω 1 2 ω( 1 2 (see [2, 9]. So we have
O Certai Subclass 151 Theorem 5. Let λ 1, α ad 1 B < A 1. If f( B (λ, α,, A, B,, the ( α 1 Au 1 1 Bu u α 1 2 du [ ( α ] 1 f( < (1 λ + λ (f ( α 2 < ( α 1 + Au 1 1 + Bu u α 1 2 du ( U, (3.1 ad iequality (3.1 is sharp, with the extremal fuctio defied by equality (3.3. Proof. From Theorem 1 we kow ( α f( (1 λ + λ (f ( α 1 + A 1 + B. Sice 1 B < A 1, we have 1 A { ( α f( 1 B < (1 λ + λ (f ( α < 1 + A 1 + B. Thus, from iequality (3.2, we ca get iequality (3.1. It is obvious that iequality (3.1 is sharp, with the extremal fuctio defied by equality (3.3. By applyig the similar method as i Theorem 5, we have Theorem 6. Let λ 1, α ad 1 A < B 1. If f( B (λ, α,, A, B,, the ( α 1 + Au 1 1 + Bu u α 1 2 du [ ( α ] 1 f( < (1 λ + λ (f ( α 2 < ( α 1 Au 1 1 Bu u α 1 2 du ( U, (3.11 ad iequality (3.11 is sharp, with the extremal fuctio defied by equality (3.3. emark 3. From Theorem 5-6 we also ca obtai the correspodig results about some other special classes of aalytic fuctios, here we do t give uecessary details ay more.
152 Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua Theorem 7. Let λ 1, α, 1 B 1, A B ad A. If f( = + k=+1 a k k B (λ, α,, A, B,, the a +1 A B ( + α(λ + 1, (3.12 ad iequality (3.12 is sharp, with the extremal fuctio defied by equality (3.3. Proof. Suppose that f( = + k=+1 a k k B (λ, α,, A, B,, the we have (1 λ f ( α ( ( f( + λ 1 + f ( (f ( α f( f ( = 1 + ( + α(λ + 1a +1 + 1 + A 1 + B. It follows from Lemma 2 that ( + α(λ + 1a +1 A B. (3.13 Thus, from (3.13, we ca get (3.12. Notice that f( = + A B ( + α(λ + 1 +1 + B (λ, α,, A, B,, we obtai that iequality (3.12 is sharp. Ackowledgmets This work was supported by the Scietific esearch Fud of Hua Provicial Educatio Departmet ad the Hua Provicial Natural Sciece Foudatio (No. 5JJ313 of the People s epublic of Chia. The authors would like to thak Professor H. M. Srivastava for his careful readig of ad costructive suggestios for the origial mauscript.
O Certai Subclass 153 efereces [1] C. Pommereke, Uivalet Fuctios, Vadehoeck ad uprecht, Göttige, 1975. [2] M.-S. Liu, O certai class of aalytic fuctios defied by differetial subordiatio, Acta Math. Sci. Ser. B Egl. Ed. 22 (22, 388-392. [3] S.-H. Li, Several iequalities about the α+iµ type of λ-bailevič fuctios of order β, Adv. Math. 33 (24, 169-173 (i Chiese. [4] S. Owa, O certai Bailevič fuctios of order β, Iterat. J. Math. Math. Sci. 15 (1992, 613-616. [5] S. Owa ad M. Nuokawa, Properties of certai aalytic fuctios, Math. Japo. 33 (1988, 577-582. [6] S. S. Miller, P. T. Mocau, ad M. O. eade, All alpha covex fuctios are uivalet ad starlike, Proc. Amer. Math. Soc. 37 (1973, 553-554. [7] S. S. Miller ad P. T. Mocau, Differetial subordiatio ad uivalet fuctios, Michiga. Math. J. 28 (1981, 157-171. [8] W. W. ogosiski, O the coefficiets of subordiatio fuctios, Proc. Lodo Math. Soc. (Ser. 2 48 (1943, 48-82. [9] Z.-G. Wag, C.-Y. Gao, ad M.-X. Liao, O certai geeralied class of o-bailevič fuctios, Acta Math. Acad. Paedagog. Nyhái (N.S. 21 (25, 147-154.