Univalent and Starlike Generalized Hypergeometric Functions

LEMMA 1. Let w(z) be regular in the unit disk °U, with w(0) = 0. Then, if \w(z) | attains its maximum value on the circle \z\ = r (0 = r < I) at a point Z], we can write (2.2) zxw'(zx) = mw(z\), where m is real and m = 1.

LEMMA 2. Every close-to-convex function is univalent. We now prove our first result on univalent generalized hypergeometric functions, contained in

THEOREM 1. Let the generalized hypergeometric function pE (z) defined by (1.5) satisfy the condition (2.3) pFq(ah , an\ b, K>z) P lia, 7 = 1 7 = 1 X for some fixed fi p ZPW*U- • .,ap;

COROLLARY 1. Let the generalized hypergeometric function FAz} defined by (1.5) satisfy the condition (2.15) for z ^ °U, where (2.4) holds true. Then pF (z) is univalent in the unit disk °U.

COROLLARY 2. Let the generalized hypergeometric function pF (z) defined by (1.5) satisfy the condition P lia, P pF^(ax,. ..,ap; bx,. ..,bq,z)- 7 = 1 q lib, 7 = 1 7 = 1 < ru, 7 = 1

LEMMA 3. Let the function f(z) defined by (1.1) satisfy the condition

THEOREM 2. Let the generalized hypergeometric function pF (z) defined by (1.5) satisfy the condition z pi^(a„. ..,ap; 6„. ..,*,; z) pFq(ax,.. .,ap; bx,.. -,bq;z) < 1 - a (z Œ <%) (3.7) forO ^ a â 1/2.

COROLLARY 3. Let the generalized hypergeometric function pF (z) defined

THEOREM 3. Let the generalized hypergeometric function F (z) defined by (1.5) satisfy the condition (3.13) zpF%(ax,. ..,ap; b\,. ..,bq,z) rtyai,.. -,ap; by,.. • , bq; z) < (1 - a) l\\ - -a + az) (z e ^ ) for 0 ^ a ^ 1/2 W (3.i4) n ^ ^ o . Then pFq(z) is s tar like of order a with respect to 1.

THEOREM 4. Let 0 < b ^ 2 and b = a < c. Also let the hypergeometric function 2F\(a, b\ c\ z) be defined by (1.5) with p = 2 and q = 1. 77* ew the function z 2F\(a-> b\ c\ z) is in the class Sf*(\ — (1/2)/?). ^ ^ a r i c 4. Since ^*(1 - (1/2)6) Ç ^ * c ^ 0 < è ë 2, the function A(z) defined by A(z) = z 2^1 (#, 6; c; z) is univalent in °ll under the hypotheses of Theorem 4. 4. Convex generalized hypergeometric functions of order a. Corre- sponding to Theorem 2 and Theorem 4, we have the following

THEOREM 5. Let the generalized hypergeometric function F(z) defined by (1.5) satisfy the condition (3.7) for 0 ^ a ^ 1/2. Then the function is in the class Jt{a).

THEOREM 6. Let 0 < b ^ 2 a^d b ^ a < c. Also let the hyper geometric function 2F\(a, b\ c; z) 6e defined by (1.5) vwY/z p = 2 and q = 1. 77zew the function z 3F2(a, b, 1; <?, 2; z) /j /'« fAe c/aw Jf(l - (1/2)6) /or z G <&

LEMMA 4. If a ^ ft ^ 1 a/id a < 1, //z^« (5.7) Sf\2 - 2^8, 2 - 2a)^*(a) c S?*(p) c ^*(a).

THEOREM 7. If the function f(z) defined by (1.1) is in the class JT(l/2), then Qxf G ^*(l/2) for 0 ^ À < 1, f/wrt w, (5.16) S2AJf(-) c W - ) (0 ^ A < 1). Proo/. With the aid of (5.5) and (5.15), we have (5.17) S2AJfj-) = J^(2, 2 - À)J^(1, 2 ) W - ) = J^(l, 2 - X)W-J. Since ^*(l/2) c y*((l/2)A) for 0 ^ (1/2)X < 1/2, (5.18) £2AJfJ-j c jgfXl, 2 - À ) W - A ) . Further, putting a = (1/2)X and /? = 1/2 in Lemma 4, we get (5.19) ^ ( 1 , 2 - À ) W - À ] C W - )

COROLLARY 4. L ^ the generalized hyper geometric function pF(z) defined by (1.5) satisfy the condition (5.20) Then z pF'q(ax,...,ap\ bu . .., bq; z) pFq(ax,..., ap\ b\,...,bq\ z) < - (z e * ) . 2 fiA{2 i>+i^V+i(ai' • v 1;/>„..., ^ , 2; z)} e y * | l ) ,

Theorem 6 and Theorem 7, we have

COROLLARY 5. Let 1 ^ a < c. Then tix{z 3F2(a, 1, 1; c, 2; z) } G W - j , where 0 = A < 1.

THEOREM 8. Let 0 â A < 2. Then (5.21) Î2 AJT(-A) = j ( - A

COROLLARY 6. Let the generalized hyper geometric function pFiz) dejined by (1.5) satisfy the condition (5.23) z pF^(al9 . . .,ap; Z>„ . . . , bq\ z) ,Fq(a]9 . . . , ap\ bl9...9bq, z) ^ 1. < 1 - - A (z e <%) forO ^ A Then (5.24) Qx{z p+lFcl+](a],. . . , ap, 1; *„ . . ., bq, 2; z) } e J ^ A ) .

COROLLARY 7. Le/ 0 ^ A < 2 <2«<i 2 — A ^ a < c. TTzen (5.25) flA{z 3F2(a, 2 - A, 1; c, 2; z) } G J | - A ) . Finally, we prove the following characterization theorem for the generalized hypergeometric function pF(z) by using the linear operator £\a9 c).

THEOREM 9. Let the generalized hypergeometric function pF(z) dejined by (1.5) satisfy the condition (3.13) for 0 = a ^ 1/2, and let the constraint (3.14) hold true. Then Fa+M + 1, p + \Âci+\y"\ ap + 1, 1; b] + 1, .6, + 1, 2;z)

LEMMA 5. Let h(z) and g(z) be analytic in the unit disk °U and satisfy h(0) = g(0) = 0, h'(0) * 0, g'(0) ¥= 0. Suppose that, for each o ( \a\ = 1) and p ( \p\ = 1), we have (6.1) h(z) * (\+ paZ)g(z) ^ 0 (z e * - {0} ). \ 1 — oz ! Then, for each function F(z) analytic in the unit disk °U and satisfying the inequality (6.2) Re{F(z) } > 0 (z e <%), (6.3) R e f ^ ^ l > 0 (z e * ) ,

THEOREM 10. Let the function f(z) defined by (I.I) be in the class £f* and let, for each a (\a\ = 1) and p ( \p\ = 1), (6.4) &(2, 2 - X)(l + pOZ f(z)\ ^ 0, Vz G qi - {0}. V 1 — az I Then Q, f(z) is also in the class Sf*.

LEMMA 6. Let the function f(z) be in the class £f*. Then (1 + \z\ f\f(z) \7,\ umi zf'(z)\ (6.7) for z Az) |z|log s 1 (1 - \z\ )log -1 - \z\ Equality in (6.7) holds true for the Koebe function

COROLLARY 8. Under the hypotheses of Theorem 10, /(l + \z\f\SlXf(z)\ (6.9) GX/(z) 1 + (1-1*1 )log( 1 + z A - \z\ for z e °U. Equality in (6.9) holds true for the function J(z) given by z (6.10) f(z) = J?(2 - X, 2)1 Z_ X \{\ - zf

LEMMA 7. Let the function j(z) be in the class Sf*. Then (6.11) Re and (6.12) Re 1 - \z\ \f(z) 1 + \z\ 2|z|log (i - \z\ ?\m (i - wW1 + lzl for z e <%. Equality in (6.11) is attained for a function of the form (6.13) f(z) = . 9e

COROLLARY 9. Under the hypotheses of Theorem 10,

THEOREM 11. Let the function j(z) defined by (1.1) he in the class JTand let, for each o ( |cr| = 1) and p ( \p\ = 1), (6.20) ^(2, 1)J^(2, 2 - X)(l + P°Z f(z)\ ^ 0, V z E f - {0}. \ 1 — oz I Then £2 f(z) is also in the class Jfc

LEMMA 8. Given JU, with — oo < [x < oo, let OO -. (6-21) / » = 2 —-~,z"+]

THEOREM 12. Let the function f (z) defined by (1.1) be in the class s^ and satisfy, for each o ( \a\ = 1) and p ( \p\ = 1), (6.22) Sf(2, 2 - X ) ( i _ L ^ ( / * / ( z ) ) ) \ 1 — oz r i for /x = 0, the function j (z) being given by (6.21). Then £2A(^ * /(z) ) is in the class 9>*.

COROLLARY 10. If f(z) is in the class tf* and satisfies the condition (6.22) for n ^ 0, then Q^if^ */(z) ) is also in the class y*,/^(z) Z?ez>2g g/v<?/7 fty (6.21).

Theorem 10. Ruscheweyh and Sheil-Small [12] (see also [2, p. 248, Theorem 8.6'] ) proved the following lemma.

LEMMA 9. Iff(z) e ^ * and g(z) <E X then f * g(z) €= ^ * .

THEOREM 13. Let the function J(z) defined by (1.1) be in the class JTand satisfy, for each o ( \o\ = 1) and p ( \p\ = 1), (6.25) J?(2, 1)^(2, 2 - \)(\+ P 0 Z(/ M */(z) )) * 0 (z G * - {0} ) \ 1 — oz 1 for /x ^ 0, the function f(z) being given by (6.21). Then Q x(/ *f(z)) is also in the class X

Definition 1. The fractional integral of order X is defined, for a function /(*), by (5.11) D7xf(z) = — f\ ^ xdÇ, where À > 0, f(z) is an analytic function in a simply-connected region of the z-plane containing the origin, and the multiplicity of (z — f) " ! is

Definition 2. The fractional derivative of order X is defined, for a function /(*), by (5-12) DXf(z) = 1 ± fl-^d^ * IX1 - X)dz J °(z - f) where 0 ^ X < 1, f(z)

Definition 3. Under the hypotheses of Definition 2, the fractional derivative of order n + À is defined by (5.13) iy!+Xf(z) = ^-n&J(z\ dz where 0 ^ X < 1, and « G J U {0}. By using these definitions of fractional calculus we introduce the linear