SOME INEQUALITIES INVOLVING BETA AND GAMMA FUNCTIONS Prnesh Kumr,S.P.Singh 2 S.S.Drgomir 3 Keywords: Bet function; Proility distriution function; Moments; Moment rtios. ABSTRACT An inequlity for the Euler s Bet -function is estlished. Proerties of the Bet roility distriutions like men, vrince, moment rtios, re considered to rove some more inequlities.. INTRODUCTION Bet roility distriutions hve estlished their usefulness in the sttisticl nlysis of reliility, life testing models in mny other lictions [Bin (978)]. A et rom vrile (r.v.) X with rmeters, hs the roility density function df fx :, x x B, ; x. where, :, B, ΓΓ Γ These distriutions ossess numer of sttisticl roerties [Johnson Kotz (97),.4; Ord (972),. 6], some of interesting eing : these re (i) memers of Person fmily, (ii) exonentil () wrt ( known), () wrt ( known), (c) wrt oth, (iii) monotone likelihood rtio (MLR) in ()T x logx,( known), () T 2 x log x, ( known), (iv) unimodl, ; not unimodl,,or,. These hve (i) incresing filure rte (IFR),, not IFR, not decresing filure rte (DFR),rx x,for, (ii) IFR,rx, x x where rx is the filure rte. The resent er ims t first estlishing n inequlity on the Bet - function then, to use moments moment rtios of the Bet rom vriles to derive some more inequlities.,2 Dertment of Mthemtics & Sttistics, Memoril University of Newfoundl,St. John s,nf,cnd AC 5S7. 3 School of Communictions Informtion, Victori University of Technology, Melourne City MC, Victori 8, Austrli..
2. AN INEQUALITY FOR THE EULER S BETA - FUNCTION We strt with the following lemm. Lemm. Let m, n,, q e ositive rel numers, such tht (-m) (q-n) (. Then, B,qBm,n B,nBm,q 2. Γ nγq m Γ qγm n 2.2 Proof. Define the mings : f, g, h : [,],, giveny fx x m,gx x q n hx x m x n 2.3 As mq n, the mings f g re the sme (oosite) monotonic on, h is non-monotonic on,. Alying the well known Ceysev s integrl inequlity for synchronous (synchronous) mings [Leedev (957)], i.e., hxdx hxfxgxdx hxfxdx hxgxdx 2.4 we cn write the inequlity x m x n dx x m x n x m x q n dx i.e., x m x n x m dx x m x n x q n dx x m x n dx x x q dx 2
x x n dx x m x q dx, y virtue of (.), the inequlity (2.) is roved. The inequlity (2.2) follows from (2.) y tking into ccount tht B,q ΓΓq 2.5 Γ q for ll,q. We shll omit the detils. The following interesting corollries my e noted s well: Corollry 2.2. Let, m.then,we hve the inequlities B,Bm,m B 2,m 2.6 Γ 2 m Γ2Γ2m 2.7 Proof. In the ove lemm, if we choose q, m n, wehve mq n m, thus, which roves the inequlity (2.6). B,Bm,m B,mBm, The inequlity (2.7) follows from (2.6) through (2.5). Corollry 2.3. Given two ositive rel numers u v, the geometric men of Γ(u) Γ(v) is greter thn the gmm of the rithmetic men of u v. This result follows y re-writing (2.7) s Γ u 2 v where u v m. ΓuΓv 3. INEQUALITIES FOR MOMENTS OF BETA RANDOM VARIABLES A distriution function determines set of moments when they exist. The first moment 3
out origin, recognized s the men or center of grvity the second moment out men, mesure of the sred or disersion of the oultion, re frequently studied rmeters of oultion. Other roerties such s skewness kurtosis re defined in terms of the higher moments. The r th moment of the r.v. X out the origin is defined y r X EX r,where E denotes the mthemticl execttion. The r th centrl moment of the r.v. X, r X, cn e derived from r r X i i r i i r X X,r,2,... Now, we resent theorem on the moments of the rom vriles which follow Bet dfs. Theorem 3.. Let the r.v. X Y e such tht X B,q Y Bm,n,,q,m,n. Further, let the r.v. U V e defined s U B,n V Bm,q. Then, for (-m)(q-n), EX r EY r EU r EV r B,nBm,q B,qBm,n,r,2,... 3. Proof. In (2.3) of lemm 2., we choose fx x m,gx x q n hx x rm x n 3.2 Then, on sustituting these mings in (2.4), we rech t the inequlity in Theorem 3.. Remrk 3.2. The inequlities for the solute moments of r.v. X, r X E X r the fctoril moments, r X EX r, out origin, my e otined from Theorem 3., on relcing r. y r. r., resectively. Similrly, corresonding inequlities for the moment genertion functions, M x t Ee tx, chrcteristic function, x t Ee itx my e esily otined from lemm 2.. An interesting result from this theorem follows s : Corollry 3.3. For q,m n, EX r EY r E r UE r V Γ2Γ2m,r,2,... 3.3 Γ 2 m 4. INEQUALITIES FOR MOMENTS OF TWO BETA RANDOM VARIABLES 4
Theorem 4.. Let the r.v. X Y e such tht X B,q Y B,n. Then, for,q,m,n EX r EY r Γ qγm n,r,2,... 4. Γ nγm q ccording s (-m)(q-n). Proof. We choose in (2.3) of lemm 2., fx x r m,gx x q n hx x m x n 4.2 Then, sustituting these mings in (2.4) results in the desired exression in Theorem 4.. Corollry 4.2. For q(), m(n), EX r Γ 2 n EY r Γ2Γ2n,r,2,... 4.3 5. INEQUALITIES FOR HARMONIC MEANS OF TWO BETA RANDOM VARIABLES Theorem 5.. Let the r.v. X Y e such tht X B,q Y B,n. Denote the hrmonic mens of r.v. X r.v. YyHM(X) E( HM(Y) E(. Then, for,q,m,n X Y, HMX HMY B,nBm,q B,qBm,n 5. ccording s (-m)(q-n). Proof. We choose in (2.3) of lemm 2., fx x m,gx x q n hx x m x n 5.2 Then, sustituting these mings in (2.4) roves Theorem 5.. Corollry 5.2. For q(), m(n), 5
HMX HMY Γ2Γ2n Γ 2 n 5.3 6. INEQUALITIES FOR VARIANCES OF TWO BETA RANDOM VARIABLES Theorem 6.. Let the r.v. X Y e such tht X B,q Y B,n. Denote the vrinces of r.v. X r.v. YyV(X) X 2 X V(Y) Y 2 Y. Then, for,q,m,n VXBm,nB,q VYBm,qB,n Bm,qB 2,n B,n Bm,nB2,q B,q 6. ccording s (-m)(q-n). Proof. We consider the inequlity in theorem 4. y choosing r rewrite. in terms of V(.).Then, we get VX XBm,nB,q VY YBm,qB,n 6.2 Now sustituting X q Y desired inequlity in Theorem 6.. n in the ove exression, we rech t the Corollry 6.2. Denoting coefficients of vrition of the r.v. X Y y CV(X) CV(Y) where CV(.) V., the inequlity for CV (X) CV(Y) follows:. CV 2 X CV 2 Y qγm nγ q nγm qγ n 6.3 ccording s (-m)(q-n). Proof. From ( 6.), we hve 6
VX X VY Y Γm nγ q Γm qγ n Y X Now, sustituting X corollry. q Y n, in the ove exression, we rove the 7. SOME MORE INEQUALITIES FOR GAMMA FUNCTIONS We note tht the men vrince of Bet r. v. Z with rmeters u v re uv, resectively. Then, we hve for Bet r.v s. X Y, defined s ove, uvuv EX q, EY n u uv VX q q q, VY n n n Using these vlues, the inequlity (4.) (6.3) yield Γ n Γm q Γ q Γm n 7. nq q qn n Γ q Γm n Γ n Γm q 7.2 ccording s (-m)(q-n), where, q, m, n. BIBLIOGRAPHY [] Bin, L.J.(978). Sttisticl nlysis of reliility life testing models, Mrcel Dekker Inc.. [2] Johnson, N.L. Kotz, S (97). Continuous univrite distriutions - II, Houghton Mifflin, Boston. [3] Leedev, H. H. (957). Secil functions their lictions, (Ed.), Tehnic, Buchrest. [4] Ord, J. K. (972). Fmilies of frequency distriutions, Hfner Pu. Co., New York. 7