H. M. Srivastava and S. Owa

THEOREM 2.1. Let the function f(z) be defined by (1.1). If (2.1) Σ nt a\ n\¥¥ n=l (1 + a)n I (C)n for a > — 1. then f(z) e f(α, c; α). TΛe result (2.1) is sharp.

COROLLARY 2.1. Let the function f(z) be defined by (1.1). // (9 #\ v ( ~^~ a' n \n < i V z °/ Z J Z . — —-T— 1 α« +1 ^ 1 w=i (1 + or)TO

THEOREM 2.2. Let the function f(z) be defined by (1.1). If (2.10) ^ ^ (2)n(2 + a)n (1 + a)n for a > — 1, then f(z) e iΓ(a, c; a). The result (2.10) is sharp.

COROLLARY 2.2. Let the function f(z) be defined by (1.1). If (2.12) f]i^ί?Jl«k (l)(l + ) /or α > — 1, then f(z)ei/ r(a). The result (2.12) is sharp for the functions given by https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

THEOREM 2.3. Let the function f(z) be defined by (1.26). Then f(z) e yo(α, c; a) if and only if (2.14) n=i (1 + a)n(c)n for a > — 1, where (a)J(c)n > 0. The result (2.14) is sharp.

COROLLARY 2.3. Let the function f(z) defined by (1.26) be in the class yo(α, c; a) with (a)J(c)n > 0. Then (2.18) |On+1 ^ ^ W |On+1| < ^ ^ W Y (2 + a)Ja)n The result (2.18) is sharp for the functions f(z) given by (2.17).

COROLLARY 2.4. Let the function f(z) be defined by (1.26). Then f(z) e i^oia) if and only if (2.19) ±^±^k\an+1\^l »=i (1 + a)n for a > — 1. The result (2.19) is sharp for the functions given by (2.20) f(z) = z - U±^kz* + 1 (n e Jί) . (2 + ά)n

Theorem 2].

THEOREM 2.4. Let the function f(z) be defined by (1.26). Then f(z) e #"0(α, c; a) if and only if (2.21) ±<&<1±}Ά (l)( «=i (l) n(l + a)n(c)n for a > — 1, where (a)J(c)n > 0. The result (2.21) is sharp.

COROLLARY 2.5. Let the function f(z) defined by (1.26) be in the class la, c; a) with (a)J(c)n > 0. Then (2)n(2 + a)n{a)n The result (2.23) is sharp for the functions f{z) given by (2.22). https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

COROLLARY 2.6. Let the function f(z) be defined by (1.26). Then f(z) e (a) if and only if /or or > — 1. The result (2.24) is sharp for the functions given by (2.25) f{z) = Z- fflrXl + *)n zn + l (Λ 6 Λθ .

Corollary 2.6 corresponds, for — 1 < a <I 0, to a result for the class tf(a) given by Silverman [14, p. Ill, Corollary 2]. §3. Distortion theorems With a view to determining the extreme points of the classes i^oia, c; a) and τΓ0(α, c; a), we first prove

LEMMA 3.1. The class i^0(a9 c; a) with (a)J(c)n > 0 is convex considered as a linear space over the field of real numbers.

THEOREM 3.1. The extreme points of the class -f~0(a,c;a) with (ά)J(c)n > 0 are the functions fn(z) (n ^> 0) given by (3.4) and (3.5). In precisely the same manner, we can establish

THEOREM 3.2. The extreme points of the class τF0(α, c; a) with (a)J{c)n > 0 are the functions fn(z) (n ^ 0) given by (3.4) and (3.12) fn{z) = z - %>% + ^ % % ^ z (ne/). (2)n(2 + a)n(a)n

THEOREM 3.3. Let the function f(z) defined by (1.26) be in the class rro(a, c; a) with (α)n/(c)n > 0 and a ^ c. Then (3.13) \z\ - %±^ i ^ %^W £ 1/(2)1 = \z\ + i ^ (2 + a)a (2 + a) for ze°tt. Furthermore, if either a ^ 0 and

COROLLARY 3.1. Let the function f(z) defined by (1.26) be in the class yo(α, c; a) with {a)J(c)n > 0 and a^c. Then the unit disk °tt is mapped onto a domain that contains the disk \w\ < r0, where r0 is given by (3.23) /•„ = l — a)c (2 + a)a + a

COROLLARY 3.2. Let the function f(z) defined by (1.26) be in the class ). Then (3.24) for zefy. Furthermore (3.25) 2 + α ' 2 + α ' ' /or zefy. The results (3.24) and (3.25) are sharp for the function (3.26) /(*) = z - (λ±£L

THEOREM 3.4. Let the function f(z) defined by (1.26) be in the class iΓ^(a, c; a) with (a)n/(c)n > 0 and a ^ c. Then /f l \ _ /-I i \ (3.27) 2(2 + a)a for zeW. Furthermore (3.28)

COROLLARY 3.3. Let the function f(z) defined by (1.26) be in the class τF0(α, c; a) with (a)J(c)n > 0 and a ^ c. Then the unit disk °U is mapped onto a domain that contains the disk \w\ < ru where rλ is given by (3.30) 7\ = 1 -

COROLLARY 3.4. Let the function f(z) defined by (1.26) be in the class a). Then for zeW. Furthermore (3.32) 1 -l\±±)\z\ ύ \f'(z)\ \2 + / ) \ \ ύ \f()\ + ( a / \2 + a for ze&. The results (3.31) and (3.32) are sharp for the function (3.33) f(z)

THEOREM 4.1. Let the function f(z) defined by (1.26) be in the class Ψ*la, c; a) with (a)J(c)n > 0. Then f(z) is starlike of order δ (0 <: δ < 1) in the disk \z\ < r2, where (4.1) r2 = infΓ (1 ~ « 2 + «^L(n + l-5Xl

COROLLARY 4.1. Let the function f(z) defined by (1.26) be in the class "Γoia). Then f(z) is starlike of order δ (0 <L δ < 1) in the disk \z\ < r8, where (4.6) r^infί V ~ M + «)* Γ. Similarly, by applying Theorem 2.4 instead of Theorem 2.3, we have

THEOREM 4.2. Let the function f(z) defined by (1.26) be in the class IT,{a, c; a) with (a)J(c)n > 0. Then f{z) is convex of order δ (0 < δ < 1) in the disk \z\ < r2, where r2 is given by (4.1).

COROLLARY 4.2. Let the function f(z) defined by (1.26) be in the class iΓ0(a). Then f(z) is convex of order δ (0 ^ δ < 1) in the disk \z\ < r3, where r3 is given by (4.6). § 5. Further applications of Theorem 2.3 and Theorem 2.4 We deduce several interesting relationships between the various sub- classes of si as further consequences of Theorem 2.3 and Theorem 2.4. We first state

THEOREM 5.1. Let (a)J(c)n > 0. Then (5.1) if la, c; a) C -rJa, c; - — ^ — ) . The result is the best possible. https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

THEOREM 5.2. Let (d)J(c)n > 0. Then, if a^c, (5.5) Πa, c;a)d <rQ(a) , and, if a < c, (5.6) τ r o ( a , c ; a ) 3 ^ o ( a ) .

THEOREM 5.3. Let (a)J(c)n > 0. Then, if a :> c, https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

THEOREM 6.1. Let the function f(z) be defined by (6.1). Then f(z) e fc(α, c; α) i/ and onZ^ if https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

COROLLARY 6.1. Let the function f(z) be defined by (6.1) with a = c. (z) e i^icia) if and only if (6.10) Σ | + 4 α n + ^ l «=*+i (1 + a)n *-i TΛe result (6.10) is sharp for the function (6.11) /(*) = 2ί - έ A ± _ ^ i - ^ + i < = l (2 + ah l-ΣPι) ^ ί

THEOREM 6.2. Let the function f(z) be defined by (6.3). Then f(z) e (a, c; a) if and only if result (6.12) is sharp for the function (6.13) f(z) = z-ΣBtPiZ i+1 ί = l (l)n(l + α)n(c) n(l-ΣA) - ^ ^ - i - z ^ 1 (n > AJ + 1) . (2)n(2 + α)n(α)n

COROLLARY 6.2. Let the function f(z) be defined by (6.3) with a = c. Then f(z) e 1^k(a) if and only if Y 1 (2)w(2 + <x)n < i _ 77&e result (6.14) is sharp for the function (6.15) /(») = ^ - Σ () ii (2) (2), (2 + αJ (2),(2 + α),

THEOREM 6.3. The extreme points of the class Ϋ"k(a, c; a) with (a)J(c)n > 0 are (6.16) fkλ and (6.17) /»+i(*) = * - Σ ' ^ k + 1) , (2 + a)n{a)n where At is given by (6.2).

THEOREM 6.4. The extreme points of the class iΓk(a9 c; a) with (a)J(c)n > 0 are (6.18) fk + 1(z) = z - Σ j (6.19) fn + 1(z) = z - Σ • ΐ = l (2)n(2 + α),(α), iϋ/iere JBJ ίs ^iuen 6^ (6.4). > k https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

LEMMA 7.1 (Jack [4]). Let w(z) be regular in the unit disk °tt with w(0) = 0. Then, if \w(z)\ attains its maximum value on the circle \z\ = r (0 <^ r < 1) at a point zQ9 we can write (7.2) ZQW(Z0) = mw(z0) , where m is real and m ^ 1. We now state our first characterization theorem involving the func- tional fβ(f).

THEOREM 7.1. Let the function f(z) defined by (1.1) by in the class y{a^ c; a). Then, for β ^ a > —1, /β{f) is also in the class i^{a, c; a).

COROLLARY 7.1. Let the function f(z) defined by (1.1) be in the class τr(a, c; a). Then, for β^a> - 1 , (7.14) js?(α, c)/β(f) e <f* and fβ(J?(a,c)f(z))e<7*.

THEOREM 7.2. Let the function f(z) defined by (1.1) be in the class τΓ(α, c; a). Then, for β 2> a > — 1, <///) is αZso in ί/ie class iΓ(a, c; α).

COROLLARY 7.2. Let the function f(z) defined by (1.1) be in the class if {a, c; a). Then, for β^a> - 1 , (7.18) &(a,c)/β(f)ejr and fβ(J?(a,c)f(z))eJT. § 8. Applications of the fractional calculus operator Ω λ From among the various definitions of fractional calculus (that is, fractional derivatives and fractional integrals) given in the literature cited, we choose to recall here the following definitions which were used recently by Owa [8] (and by Srivastava and Owa [17]): DEFINITION 8.1. The fr

THEOREM 8.1. Let the function f(z) defined by (1.26) be in the class -To(2, 2 - i; a) for λ < 1. Then (8.9) nwwi (8Λ0) /or ^ef. TΛe reswte (8.9) and (8.10) are

THEOREM 8.2. Let the function f(z) defined by (1.26) be in the class τΓ0(2, 2 - λ; a) for λ < 1. Then and /or z e t . ΓΛβ results (8.13) αrcd (8.14) are sharp for the function (8.15) f(z) = 2 - (1 + «X2 ~ i) g 2 . 4(2 + a )

THEOREM 8.3. Let λ<l. Then (8.16) £

COROLLARY 8.1. Let λ < 1. Then (8.19) Ω 1+iiΓ0(2, 2-λ;a)d Πa) .

COROLLARY 8.2. Let λ<l. Then (8.20) Λ I+1#"t(2, 2 - a α) drT^a). Next we prove https://doi.org/10.1017/S0027763000000854 Published online by Cambridge University Press

THEOREM 8.4. Let λ < 1. Then (8.21) Ω λiΓ(l, 2 - λ; a) C y(α) . Proo/. Since f(z) e 7Γ(1, 2 - λ; a) implies that f(z) e J£?(2 - Λ, we have (8.22) β'τr(l, 2-λ\a)<Z = JSf(2, 2 - = JSf(2, l)τT(α) - τT(α) ,

COROLLARY 8.3. Let λ < 1. jΓ/ιeτι (8.23) Ω λ1T0(l, 2-λ;a)d rT0(a) .

COROLLARY 8.4. Let λ < 1. Then (8.24) β'τr*(l, 2 - λ; a) C ^ Λ ( α ) . Finally, we prove the following theorem involving generalized hyper- geometric functions.

THEOREM 8.5. Let λ < l . Then (8.25) zq+2Fq+ί(2, , 2, 2; 1, . . ., 1, 2 - λ; z) e β« + 1 + i(^*(i) Π ^) .