Some inequalities involving the Fox's H-function

Size: px

Start display at page:

Download "Some inequalities involving the Fox's H-function"

Alicia Oliver
5 years ago
Views:

1 Proc. Indian Acad. Sci., Vol. 83 A, No. 1, 1976, pp Some inequalities involving the Fox's H-function R: K. RAINA Department of Mathematics, University of Udaipur, S.K.N. Agriculture College, Jobner (Rajasthan) C. L. KouL Department of Mathematics, M.R. Engineering College, Jaipur-4 MS received 21 May 1975 ABSTRACT In this paper we establish certain inequalities for the Fox's H-function with the help of certain known inequalities for the generalized hypergeometric function. 1. INTRODUCTION LUKE obtained a number of inequalities for the generalized hypergeometric function pfq in the case p = q or p = q + 1, with the help of some well-known Pade's approximations for a particular form of the Gaussian hypergeometric function 2F1 one of whose numerator parameters is unity. Carlsonz also developed certain inequalities for a hypergeometric function of n-variables. The aim of this paper is to derive certain inequalities for the Fox's H-function with the help of certain known inequalities established by Luke. The Fox's H-function is defined and represented in the following manner: M w p H-P(bi #;s)17r(1 aj + ajs) m n (a3 aa)l,p l. 1 t-1 r-t H (bj,gjzj ZWi a p zsc's II r(1 b^+fljs) II r(al als) C (1.1) z 0, m, n, p, q are integers satisfying 1 <m < q, 0 < n < p, al (j = l, 2,..,, p), fil (j =1, 2,..., q) are positive integers and aj (j = 1,..., p) b (j = 1, 2,..., q) are complex numbers. The contour C is a straight line running from -o ioo to-a+ loom such a manner that the poles of r (b3 pis) A-3 Jan

2 R. X. RAINA AND C. L. KouL j =1,..., m lie to the right and all the poles of I' (1 aj + ajs) j =1,..., n lie to the left of the contour. All the poles are assumed to be simple. The integral in (1.1) converges when (i) A=E(ai) '(ai)+2'(fi) P ii L'(fli)> 0. 1 '1+1 1 m+l (ii)i arg z I < Z ATr. The function defined in (1.1) represents an analytic function if: (iii) z : o, S =() E (al) >0 or (iv) 0, 0<Iz^<D, D = n (al )- ' in (91)Pi For the behaviour of the H-function for large values of z, we refer to Braaksma THE INEQuALiTiEs The following inequalities involving the Fox's H-function are established: For z> 0, p> 0, P,?a, >0 (j =1,2,...,P), 8 =ap = p + l ='n ' r (a9)' we have (c,o)" H,2 rz a ( I (1 -- a, P) l L (0, 1), (1 A, P)] p+^ z(1 A, P), (P9 A, P)i p l < r ^Q) Ho+i.P+2. [ I ( 0, 1), (a,1,p) (a9 l

3 Some inequalities involving the Fox's H-function 35 20a (;VP) Z-a^P + Qo ((a + 1) \-r_^ < (1 (o + 1) ^! r ( (a + 1) 0 -P (1 A, P) xhl;z[z\ 2 ) (0, 1),(1 A,p)] l 0<a<1, Re(A >0; (2.1) 0-)1 H `:1 (1 A, P) [ Z(0)-,O l (0, 1), (a A, P)] (1 <H +2:o+z Cz I ^A, P), (fl A, P)^,p ] (0, 1), (a A, P), (dl A, P)i,p 1'( r (Alp) XIP 9 HY 1 ( 1 A, P) l <(1 B) p z + z [z1, ( 0, 1 ), (a ^1, P)] (2.2) a>0, Re(A)>0; r +1 r(a/p) Z -a^p ao ( 0l C P ) Z -(X+1)/p a0 1 P l 2/ p 2Z'(1 ++a) ( A, P) x H;:z [z (0, 1),* (a A, P)] 0 v+2'1 I (1 A,p), (Pi A, P)^.P < j(q) Hn+i,v+z [z I ( 0, 1), (v A,p), (a.7 A, P)i,pJ r (Alp) yin ao -(X-4-1) p r (Q + 1) t 2) (^ l-p ( A, P) l X H^.2 L z \2/ (0, 1), (a A, P)J a> o, Re (A) > o; (2.3) _! (^lp) z _X,r + a a pa a (a+1) 0 X (1 A, P) X H [Z( Q + 1) 1 (0, 1), (1 A, P)]

4 36 R. K. RAINA AND C. L. Kout < R (a) H a+?:v+s ' [z, (1 A, P), (if A, P), ($$ A, P)i,P (0, 1), (1 A, P), (aj A, P)i,p < {1 ) pip) -xíp + 0 ()?¼ x H I (0, 1), ( 1 ^ A, P)J (2.4) 0>0, Re(A)>O; (1 ^, P) B-a ] H11 rz (B)-a (0, 1) _! n+^ f (1 A,P), (f1 A,P)1,P < Hn7i: fl+i Lz 1 (0, 1), (aj A, P)i,P I C (1 r B) V (Alp) z" I P ',.^,. 8 H; t z (1 (0,1) P)1 Re(A)>0; (2.5) 0-7 (QB)-,'' H2:z [z (ab)-p {1 A, P), ( 1 _ j.c, 1) (0, 1), ( 1 A, P) x HH.:::'+2 [ (1 A, P),(1 µ, 1),(1e Z (0, 1), (Q A, P), (a7 ' A, P)I,P 2 ob E(AJP) r(p 1LP) z_.rtp + 2 vb } P -- (o+1)0 1)to-f `- A, P), (1 x 2 ) Hz2 z t 2 ) f (0, 1), (1 A, P) (2.6) 0<Q<1, µ >0, 0<Re(A)<µp; A, P), (1 -` P,, 1) B Hr.^ [ (B)-a (1 (0, 1), (a A, P) I n+:. s ri z (1 A, P), (1 p, 1), (13j ^, P)^.p < H v+l, p+z L (0, 1), (ff A, p), (aj A, P)a,

5 Some inequalities involving the Fox's H-function 37 < (1 0).r(a) r (A/P) r (u. A/p) r a^^ p + 9H [ z ' (1 A, P), (1 p1) (0, 1), (a A, P) j a>0, µ>0, 0< Re(A)<µp; (2.7) r (A/P) r (,^ SIP) z_xip ao (1 0/2) r P r ^` P p p X z-c^+1ia> a0 H 2.2 rz ( A. P), ( 1!L, 1) 2 r (a + 1) l i (o, l), (t A, P) ] 0 n+2 2 [ I (1 A, P), (I µ, 1); (pi A, P)1, p j < r (Q) H.+2:.+2 L z (0.1), (a A, P), (aj A, P)1, p < r (A/P) F (µ A/P) Z_XIF _ ae _^ -(X+1) p r (1 + a) (2y -a x Hl:, z ( A, p) (i th, 1) (2.8) Y z [ ()) (0,1), (a A, P) I a>o, µ>0, 0<Re(A)<µp 1, r (A/P) r (IL A/P) -1,a Q + 1 G B pa Z +C )+1/ x H2:2 e P (1= A, P), (1 p, 1) [Z( 1 (0, 1), (1 A, P) ] < r (Q) rfi (1,1), (A, P), (1 + A Hv'+2: n+s f1 aj P)i,p j (A, P),, 1),( 1 +A a, P), ( 1 +A fj,p)1,^ C ) p ^Ca) < 1 0 ) (µ A/P) r (A/P) Z-aIp + e -^ x H2;2 [z(v I (I A, P), (1 p, 1) 1 (2.9) / (0, 1), (1.1, P) J a>0, p.>0, 0<Re(A)<,up;

6 38 R. K. RAINA AKD C. L. KoUL &-x Hi:sPOP ( 1, 1) (, p), ^,1)J (A s p+l [ 1 ( 1, 1), (1 +.1 aj, P)i, p 1 <'A H^+^. P+Z z (A, P), (A, 1), ( 1 + A flj, Ph, p J < (1 B) r (A/P)r (µ AIP ) zx/p+eh2 [ a > 0, µ > 0, 0 < Re (A) <µp. P Z (A, (1,1) P), (µ, 1) 1 (2.10) 3. RESULTS REQUIRED (a) The following inequalities due to Luke [(4.10), (4.20), (4.22), (5. 1), (5.5)] a! a required in the sequel: For z > 0, 0 flp, flj? aj > 0, (j=1,2,..., P) (i) (1 + aoz) -i < p+1.fp [a' a1 l z ] < 1 (o -I- 0) 2a0 1+- z] -,0<a<1; (3.1) (ii) ( 1 + 0z)- < p+,f'p (1 + z)-, a> 0; pp I z) < (3.2) (iii)1 a0( 1 Z)z 2(1 +z ) +1 <p+1rp \ / a,a ^ z ) aqz < I qs +i a > 0; (3.3) \1+2z1 (iv) a + \ a a1!\1+a+1/ <p+1f 1(a, Pp I z) < (1+ z) a>0, (3.4)

7 Some inequalities involving the Fox's H-function 39 (V) a OZ < pfp (ap I z) < Oe-z. (3.5) (b) The following integrals which are particular cases of a known result due to Gupta and Jain4 are also needed: (i) f t x-1 e-at (1 + bt)-µ dt b-^ H [a (b)-p (1.1, P) 1 o I' (I) 1'" (0, 1), (µ A, P) J a >0, b>0, p>0,µ>0, Re(A)>0; (3.6) (ii) f tx-1 e-qt PFq A.(13q t) dt Q n1'(^^) 1.v+^ [ 1I( 1, 1),( 1 +A a;, P)i,p1, Ha+i. a+l (A,p), H 1'(a,) (1 + A fly, P)i, q J j1 a> 0, p> 0, Re (A) > 0 ; (3.7) (iii) f tx1 (1 + at)-a (1 + bt 5)-P dt 0 a 22 [b(a)-5 ^ (' a, ^),(1 P,1) l r(a)j'(p) H2,2 (0, 1), (a A,8) J a>0,b>0,a>0,9>0, 8 >0,0< Re(A) <a+p8; (3.8) (iv) St'' (1 + bt$)-a pfq(^g1 t) dt 4. PROOFS C 1 rl I' (p1) H2. n+^ 1 (1, 1), (1 + A a9, S)i, P r () n r (a;) P+I. + 2 Lb I (A, s), (IL, 1), (1 + fl, s)1, a^ f=i (3.9) b > 0,3>0,p.>0,Re(A)>0,Re(A aj)<µ8(j=1,2,...,p) To prove (2.1), we first replace z by t in (3.1), multiply throughout by ta-1 e-zt)' and integrate with respect to t between the limits 0 to 00 and then use (3.6) and (3.7) to establish the inequality given in (2. 1). The inequalities (2.2) to (2.5) are established in a similar manner by using respectively the results (3.2) to (3.5) instead of (3. 1).

8 40 R. K. RAINA AND C. L. KouL The inequalities (2.6) to (2. 10) are established by proceeding on similar lines and by taking the multiplying factor as to-1 (1, + ztp) and using (3.8) and (3.9). Particular cases: If we put p =1 in (2.1), (2.2) and (2.3), we arrive at the inequalities obtained by Luke [(5.9), (5.17), (5.18)] after a little simplification. Also, certain new inequalities involving the Meijer's G-function can be obtained from the inequalities (2.4) to (2.10) by putting p =1 therein. However we do not record these here. REFERENCES 1. Luke, Y. L., Inequalities for generalized Hypergeometric functions. Jr. Approx. Theory 5 (2), (1972). 2. Carlson, B. C., Some inequalities for Hypergeometric functions. Proc. Am. Math. Soc. 17, (1966). 3. Braaksma, B. J. L., Asymptotic expansions and analytic continuations for a class of Barnes integrals. Compositio. Maths. 15, 278 (1963). 4. Gupta, K. C. and Jain, U. C., The H-function-II. Proc. Nat. Acad. Sci., India, 36, 601 (1966)

On some Summation Formulae for the I-Function of Two Variables

On some Summation Formulae for the I-Function of Two Variables Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 932-9466 Vol. 9, Issue (June 204), pp. 362-370 Applications and Applied Mathematics: An International Journal (AAM) On some Summation Formulae