THE EXTENDED MEAN VALUES: DEFINITION, PROPERTIES, MONOTONICITIES, COMPARISON, CONVEXITIES, GENERALIZATIONS, AND APPLICATIONS

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1 THE EXTENDED MEAN VALUES: DEFINITION, PROPERTIES, MONOTONICITIES, COMPARISON, CONVEXITIES, GENERALIZATIONS, AND APPLICATIONS FENG QI Abstrct. The etended men vlues Er, s;, y) ply n importnt role in theory of men vlues nd theory of inequlities, nd even in the whole mthemtics, since mny norms in mthemtics re lwys mens. Its study is not only interesting but importnt, both becuse most of the two-vrible men vlues re specil cses of Er, s;, y), nd becuse it is chllenging to study function whose formultion is so indeterminte. In this epositive rticle, we summrize the recent min results bout study of Er, s;, y), including definition, bsic properties, monotonicities, comprison, logrithmic conveities, Schur-conveities, generliztions of concepts of men vlues, pplictions to quntum, to theory of specil functions, to estblishment of Steffensen pirs, nd to generliztion of Hermite-Hdmrd s inequlity. 1. Definition nd epressions of the etended men vlues The histories of men vlues nd inequlities re long [9]. The men vlues re relted to the Men Vlue Theorems for derivtive or for integrl, which re the bridge between the locl nd globl properties of functions. The rithmetic-mengeometric-men inequlity is probbly the most importnt inequlity, nd certinly keystone of the theory of inequlities [2]. Inequlities of men vlues re one of the min prts of theory of inequlities, they hve eplicit geometric menings [14]. The theory of men vlues plys n importnt role in the whole mthemtics, since mny norms in mthemtics re lwys mens Definition of the etended men vlues. In 1975, the etended men vlues Er, s;, y) were defined in [51] by K. B. Stolrsky s follows [ r Er, s;, y) = s ys s ] 1/s r) y r r, rsr s) y) 0; 1.1) Dte: November 18, 2001; revised on December 13, Mthemtics Subject Clssifiction. 05A19, 26A48, 26A51, 26B25, 26D07, 26D10, 26D15, 26D20, 33B20, 41A55, 44A10, 60E15. Key words nd phrses. Etended men vlues, generlized weighted men vlues, generlized bstrcted men vlues, logrithmic conve, Schur-conve, monotonicity, comprison, definition, recurrence formul, integrl epression, gmm function, incomplete gmm function, Steffensen pirs, Hermite-Hdmrd inequlity, bsolutely completely, regulrly) monotonic conve) function, rithmetic men of function, quntum, Bernoulli s numbers, Bernoulli s polynomils. The uthor ws supported in prt by NSF # ) of Chin, SF for the Prominent Youth of Henn Province, SF of Henn Innovtion Tlents t Universities, NSF of Henn Province # ), SF for Pure Reserch of Nturl Science of the Eduction Deprtment of Henn Province # ), Doctor Fund of Jiozuo Institute of Technology, Chin. The originl mnuscript is seminr report giving t the RGMIA on December 10,

2 2 F. QI [ 1 Er, 0;, y) = r y r r ] 1/r, r y) 0; 1.2) ln y ln Er, r;, y) = 1 [ ] r 1/ r y r ), r y) 0; 1.3) e 1/r y yr E0, 0;, y) = y, y; 1.4) Er, s;, ) =, = y; where, y > 0 nd r, s R. It is esy to see tht the etended men vlues Er, s;, y) re continuous on the domin {r, s;, y) r, s R;, y > 0}. They re of symmetry between r nd s nd between nd y. Mny bsic properties hd been reserched by E. B. Lech nd M. C. Sholnder in [19] in 1970 s. Mny men vlues with two vribles re specil cses of E, for emples, Er, 2r;, y) = M r, y), power mens or Hölder mens) 1.5) E1, p;, y) = S p, y), etended logrithmic mens) 1.6) E1, 1;, y) = I, y), identric or eponentil men) 1.7) E1, 2;, y) = A, y), rithmtic men) 1.8) E0, 0;, y) = G, y), geometric men) 1.9) E 2, 1;, y) = H, y), hrmonic men) 1.10) E0, 1;, y) = L, y). logrithmic men) 1.11) Study of Er, s;, y) is not only interesting but importnt, both becuse most of the two-vrible men vlues re specil cses of Er, s;, y), nd becuse it is chllenging to study function whose formultion is so indeterminte [26] Integrl epressions of the etended men vlues. Let y t t ), t 0; gt) gt;, y) = t ln y ln, t = 0. Define function U n ; t) such tht 1.12) U 0 ; t) = t, U n+1 ; t) = U n; t) 1.13) n + 1)U n ; t) for n being nonnegtive integer nd t > 0. The direct clcultion of the n-th order derivtive of gt) for n N is complicted. However, it is esy to see tht g n) t) = y ln u) n u t 1 du, y > > 0, n N. 1.14) Recently, new epression for the i-th order derivtive of gt;, y) with respect to the vrible t ws obtined by the uthor s follows 1) i g i) t) = Γi + 1, t ln y) Γi + 1, t ln ) t i+1, 1.15)

3 THE EXTENDED MEAN VALUES 3 where i is nonnegtive integer, nd Γz, ) denotes the incomplete gmm function defined for Re z > 0 by Γz, ) = t z 1 e t dt. 1.16) The epressions 1.12), 1.14), nd 1.15) of gt;, y) look like simple, but they re importnt for us. The epression 1.14) cn be used to rewrite the etended men vlues s ) 1/s r) gs;, y) Er, s;, y) =, r s) y) 0; 1.17) Er, r;, y) = ep gr;, y) gr r;, y) gr;, y) Tking logrithm in 1.17) nd 1.18) yields 1 s gt;, y) s r ln Er, s;, y) = r t gr;, y) r ), y) ) 1 dt, r s) y) 0; gt;, y) ) gr;, y), r = s, y 0. Note tht, the integrl epressions 1.14), 1.17) nd 1.18) of the function g nd the etended men vlues Er, s;, y) ply key roles in our sequent contents Inequlities nd recurrence formule for gt;, y). Using Chebysheff s integrl inequlity, Hermite-Hdmrd s inequlity for conve functions nd the mthemticl induction, some reltionships between g) nd U n, t) re deduced, nd some recurrence formule nd inequlities of them re given. For emples Theorem 1.1 [46]). The function g) stisfies g n) ) = U n; b) U n ; ) n+1, 1.20) U n, t) = n+1 ln t) n t 1. t 1.21) Theorem 1.2 [46]). The function g+γ) g) is incresing or decresing) in for ] 1/t, γ > 0 or γ < 0). And t 0, is incresing with t. [ g+t) g) Theorem 1.3 [46]). The function g) is bsolutely nd regulrly monotonic on R for > 1, or on 0, ) for b > 1 > 1, completely nd regulrly monotonic on R for 0 < < b < 1, or on, 0) for 1 < b < 1. Furthermore, g) is bsolutely conve on R. Theorem 1.4 [46]). For k, i, j N, we hve g 2i+k)+1) g 2j+k)+1) < g 2k) g 2i+j+k+1)). 1.22) The rtio g2j+k)+1) ) g 2k) ) is incresing in. For completeness, we list definition of bsolutely regulrly, completely) monotonic conve) function s follows. Definition 1.1. A function ft) is sid to be bsolutely monotonic on, b) if it hs derivtives of ll orders nd f k) t) 0, t, b), k N.

4 4 F. QI Definition 1.2. A function ft) is sid to be completely monotonic on, b) if it hs derivtives of ll orders nd 1) k f k) t) 0, t, b), k N. Definition 1.3. A function ft) is sid to be bsolutely conve on, b) if it hs derivtives of ll orders nd f 2k) t) 0, t, b), k N. Definition 1.4. A function ft) is sid to be regulrly monotonic if it nd its derivtives of ll orders hve constnt sign + or ; not ll the sme) on, b). The bsolutely completely, regulrly) monotonic conve) functions re useful in Lplce trnsform [52]. 2. Monotonicities of the etended men vlues While studying function, we lwys consider its monotonicity t first. The etended men vlues Er, s;, y) re incresing with respect to its ll vribles. Tht is Theorem 2.1. The etended men vlues Er, s;, y) is incresing in both nd y nd in both r nd s. This theorem ws verified by E. B. Lech nd M. C. Sholnder in [20]. Lter, using epression 1.17) nd 1.18), monotonicity of the rithmetic men of function, Chebysheff s integrl inequlity, Cuchy-Schwrz-Bunikowski s inequlity nd other nlytic technique, some simple nd new proofs for monotonicity of the etended men vlues re provided in [15, 42, 44, 47]. 3. Comprison of the etended men vlues The comprison of the etended men vlues Er, s;, y) is difficult problem. It ws reseched in [20]. Five yers lter, more generl results were obtined by Z. Páles in [26]. It is restted in [25, 29] s follows. Theorem 3.1 [20, 26]). Let r, s, u, v be rel numbers with r s nd u v, then the inequlity Er, s;, b) Eu, v;, b) 3.1) is stisfied for ll, b > 0 if nd only if where r + s u + v nd er, s) eu, v), 3.2) y e, y) = ln for y > 0 nd y, y 0 for y = 0 if either 0 min{r, s, u, v} or m{r, s, u, v} 0, or e, y) = y y if min{r, s, u, v} < 0 < m{r, s, u, v}. 3.3) for, y R nd y 3.4) 4. Conveities of the etended men vlues After considering the monotonicity nd comprison, it is nturl to investigte the conveities of the etended men vlues Er, s;, y).

5 THE EXTENDED MEAN VALUES Definitions of conveities. The concepts of conveities of functions re mnifold, for instnce, the logrithmiclly conve nd the Schur-conve. Definition 4.1 [24]). A positive function f defined on n intervl I is logrithmiclly conve concve) if its logrithm ln f is conve concve). Definition 4.2 [6, 28]). A function f with n rguments on I n is Schur-conve on I n if f) fy) for ech two n-tuples = 1,..., n ) nd y = y 1,..., y n ) in I n such tht y holds, where I is n intervl with nonempty interior. The reltionship of mjoriztion y mens tht k [i] i=1 k y [i], i=1 n [i] = i=1 n y [i], 4.1) where 1 k n 1 nd [i] denotes the ith lrgest component in. A function f is Schur-concve if nd only if f is Schur-conve Conveity of the rithmetic men of function. The conveities of the weighted) rithmetic men of function integrl rithmetic men) re importnt to our proofs for conveities of the etended men vlues Er, s;, y). The following results cn be verified esily. Lemm 4.1 [47]). If ft) is n incresing integrble function on I, then the rithmetic men of function ft), 1 s ft)dt, r s, φr, s) = s r r 4.2) fr), r = s, is lso incresing with both r nd s on I. If f is twice-differentible conve function, then the function φr, s) is lso conve with both r nd s on I. i=1 In [6], N. Elezović nd J. Pečrić proved the following Lemm 4.2. Let f be continuous function on I. Then the integrl rithmetic men, 1 v ft)dt, u v, φu, v) = v u u 4.3) fr), u = v, is Schur-conve Schur-concve) on I 2 if nd only if f is conve concve) on I. The following necessry nd sufficient condition is well-known. Lemm 4.3 [6] nd [28, p. 333]). A continuously differentible function f on I 2 where I being n open intervl) is Schur-conve if nd only if it is symmetric nd stisfies tht f y f ) y ) > 0 for ll, y I, y. 4.4) Using Lemm 4.3, we cn obtin the Schur-conveities of the weighted rithmtic men of function nd the etended men vlues Er, s;, y) with, y) for fied r, s).

6 6 F. QI Lemm 4.4 [45]). Let f be continuous function on I, let p be positive continuous weight on I. Then the weighted rithmtic men of function f with weight p defined by y pt)ft)dt y F, y) = pt)dt, y, 4.5) f), = y is Schur-conve Schur-concve) on I 2 if nd only if inequlity y pt)ft)dt p)f) + py)fy) y pt)dt p) + py) holds reverses) for ll, y I. 4.6) 4.3. Logrithmic conveity of the etended men vlues. By formul 1.19) nd Lemm 4.1, we cn see tht, in order to prove the logrithmic conveity of the etended men vlues Er, s;, y), it suffices to verify the conveity of function g t) gt) g tt;, y) gt;, y) 1 gt;, y) t gt;, y) 4.7) with respect to t. Strightforwrd computtion results in g ) t) = g t)gt) [g t)] 2 gt) g 2, 4.8) t) g ) t) = g2 t)g t) 3gt)g t)g t) + 2[g t)] 3 gt) g ) t) By long intricte nd stndrd rgument, we obtin the following Proposition 4.1 [32]). If y > = 1, then, for t 0, we hve g 2 t; 1, y)g t t; 1, y) 3gt; 1, y)g tt; 1, y)g t t; 1, y) + 2[g tt; 1, y)] ) The combintion of Proposition 4.1 with equlity 4.9) proves tht g t t;1,y) gt;1,y) is concve on [0, ) with t for fied y > = 1. Thus, it follows tht the etended men vlues Er, s; 1, y) re logrithmiclly concve on [0, ) with respect to either r or s for y > = 1. By stndrd rguments, we obtin Er, s;, y) = E r, s; 1, y ), 4.11) y E r, s;, y) = Er, s;, y). 4.12) Hence, Er, s;, y) re logrithmiclly concve on [0, ) with either r or s nd logrithmiclly conve on, 0] in either r or s, respectively. Tht is Theorem 4.1 [32]). For ll fied, y > 0 nd s [0, ) or r [0, ), respectively), the etended men vlues Er, s;, y) re logrithmiclly concve in r or in s, respectively) on [0, ); For ll fied, y > 0 nd s, 0] or r, 0], respectively), the etended men vlues Er, s;, y) re logrithmiclly conve in r or in s, respectively) on, 0] Schur-conveity of the etended men vlues. The Shur-conveities re prted into two cses: conveities with respect to r, s) nd, y), respectively.

7 THE EXTENDED MEAN VALUES By the sme procedure s proof of the logrithmic conveity of Er, s;, y) nd using Lemm 4.2, we obtin the following Theorem 4.2 [35]). For fied, y > 0 nd y, the etended men vlues Er, s;, y) re Schur-concve on R 2 + nd Schur-conve on R 2 with r, s), where R 2 + nd R 2 denote [0, ) [0, ) nd, 0], 0], the first nd third qudrnts, respectively. Tking r 1, s 1 ) = 0, 2r) nd r 2, s 2 ) = r, r) for r 0, s direct consequence of Theorem 4.2, we obtin n inequlity between the generlized logrithmic men vlues defined by 1.2) nd the generlized identity eponentil) men vlues defined by 1.3) s follows Corollry [35]). Let, y > 0 nd y. Then, for r > 0, we hve [ 1 2r y2r 2r ] 1/2r) 1 ) r 1/ r y r ). 4.13) ln y ln e 1/r y yr For r < 0, inequlity 4.13) reverses The conveities with respect to vribles nd y re not much perfect. From Lemm 4.4, using the following Theorem 4.4 bout inequlities of the rithmetic men, hrmonic men nd logrithmic men, we hve Theorem 4.3 [45]). For fied point r, s) such tht r, s 0, 3 2 ) or r, s 0, 1], resp.), the etended men vlues Er, s;, y) is Schur-concve or Schur-conve, resp.) with, y) on the domin 0, ) 0, ). As by-products, some inequlities of men vlues were estblished. Theorem 4.4 [45]). Let > 0 nd y > 0 be positive rel numbers nd r R. 1) If r 0, then L r, y r ) [G, y)] r A, y)h r 1, y r 1 ), 4.14) the equlities in 4.14) hold only if = y or r = 0. 2) If r 3 2, we hve L r, y r ) A, y)h r 1, y r 1 ), 4.15) the equlity in 4.15) holds only if = y. 3) If r 0, 1], inequlity 4.15) reverses without equlity unless = y. 4) Otherwise, the vlidity of inequlity 4.15) my not be certin. The results of Theorem 4.4 implies inequlities between the etended men vlues nd the generlized weighted men of positive sequence. Theorem 4.5 [45]). Let, y > 0. Then 1) if r, s 0, 1], we hve Er, s;, y) M 2 1, 1);, y); r 1, s 1), 4.16) where M 2 1, 1);, y); r 1, s 1) denotes the generlized weighted men of positive sequence, y) with two prmeters r 1 nd s 1 nd constnt weight 1, 1) defined in Definition 5.2; 2) if r, s 0, 3 2 ), inequlity 4.16) reverses; 3) otherwise, the vlidity of inequlity 4.16) my not be certin.

8 8 F. QI 5. Generliztions of men vlues From 1.14), it is cler tht the etended men vlues cn be rewritten s y 1/s r) Er, s;, y) = ts 1 y. 5.1) dt) tr Generlized weighted men vlues. One of generliztions of men vlues, the generlized weighted men vlues M p,f r, s;, y), re clssified into two cses Continuous cse. It is nturl to generlize the concept of the etended men vlues Er, s;, y) through replcing the function t by positive function ft) nd considering weight in the integrnds in 5.1). Definition 5.1 [31, 34]). Let, y, r, s R, nd pu) 0 be nonnegtive nd integrble function, fu) positive nd integrble function on the intervl between nd y. The generlized men vlues, with weight pu) nd two prmeters r nd s, is defined by y M p,f r, s;, y) = pu)f 1/s r) s y u)du) pu)f, r s) y) 0; 5.2) r M p,f r, r;, y) = ep M p,f r, 0;, y) = M p,f 0, 0;, y) = ep M p,f r, s;, ) = f). y pu)f ) r u) ln fu)du y pu)f, r y) 0; 5.3) r u)du y pu)f ) 1/r r u)du y pu)du, r y) 0; 5.4) y ) pu) ln fu)du y pu)du, y 0; 5.5) The following lemm is clled the revised Cuchy s men vlues theorem in integrl form. Lemm 5.1 [31, 34, 47]). Suppose tht ft) nd gt) 0 re integrble on [, b] nd the rtio ft) gt) hs finitely mny removble discontinuity points. Then there eists t lest one point θ, b) such tht b ft)dt b gt)dt = lim t θ ft) gt). 5.6) Using Lemm 5.1, the bsic properties of the generlized weighted men vlues M p,f r, s;, y) were yielded s follows. Theorem 5.1 [31]). M p,f r, s;, y) hve the following properties m M p,f r, s;, y) M, 5.7) M p,f r, s;, y) = M p,f r, s; y, ) = M p,f s, r;, y), 5.8) M s r p,f r, s) = M s t p,f t r t, s)m r, t), 5.9) where m = inf fu), M = sup fu). In [31] nd [44], the monotonicity with nd y of M p,f r, s;, y) ws proved by three pproches. p,f

9 THE EXTENDED MEAN VALUES 9 Theorem 5.2. Let pu) 0 be nonnegtive nd continuous function, fu) positive, incresing or decresing, respectively) nd continuous function. Then M p,f r, s;, y) increses or decreses, respectively) with respect to either or y. Using Cuchy-Schwrz-Bunikowski s inequlity, we proved monotonicity of the generlized weighted men vlues M p,f r, s;, y) with r, s) s follows. Theorem 5.3 [48]). The generlized weighted men vlues M p,f r, s;, y) re incresing with both r nd s for ny continuous nonnegtive weight p nd continuous positive function f. Using Tchebysheff s integrl inequlity, we hve the following two theorems. Theorem 5.4 [31]). Let p 1 u) 0 nd p 2 u) 0 be nonnegtive nd integrble functions on the intervl between nd y, fu) positive nd integrble function, the rtio p1u) p 2u) p1u) n integrble function, p 2u) nd fu) both incresing or both decresing. Then M p1,f r, s;, y) M p2,f r, s;, y) 5.10) If one of the functions of fu) or p1u) p 2u) is nonincresing nd the other nondecresing, then inequlity 5.10) is reversed. Theorem 5.5 [31]). Let pu) 0 be nonnegtive nd integrble function, nd f 1 u) nd u) positive nd integrble functions on the intervl between nd y. If the rtio f1u) f nd u) 2u) re integrble nd both incresing or both decresing, then M p,f1 r, s;, y) M p,f2 r, s;, y) 5.11) holds for r, s 0 or r 0 s, nd f1u) u) 1. The inequlity 5.11) is reversed for r, s 0 or s 0 r, nd f1u) f 1. 2u) If one of the functions of u) or f1u) u) is nonincresing nd the other nondecresing, then inequlity 5.11) is vlid for r, s 0 or s 0 r, nd f1u) f 1; 2u) the inequlity 5.11) reverses for r, s 0 or r 0 s, nd f1u) f 1. 2u) Discrete cse. The discrete nlogue of the generlized weighted men vlues, the generlized weighted men of positive sequence = 1,, n ), ws defined in [30] by Definition 5.2. For positive sequence = 1,, n ) with i > 0 nd positive weight p = p 1,, p n ) with p i > 0 for 1 i n, the generlized weighted men of positive sequence with two prmeters r nd s is defined s n i=1 p i r ) 1/r s) i n M n p; ; r, s) = i=1 p i s, r s 0; i n i=1 ep p i r i ln ) 5.12) i n i=1 p i r, r s = 0. i Remrk 5.1. For s = 0 we obtin the weighted men M [r] n ; p) of order r see [24]); for s = 0, r = 1, the weighted hrmonic men; for s = 0, r = 0, the weighted geometric men; nd for s = 0, r = 1, the weighted rithmtic men. The men M n p; ; r, s) hs some bsic properties similr to those of M p,f r, s;, y), for instnce

10 10 F. QI Theorem 5.6 [30]). The men M n p; ; r, s) is continuous function with respect to r, s) R 2 nd hs the following properties M s r n m M n p; ; r, s) M, M n p; ; r, s) = M n p; ; s, r), p; ; r, s) = M s t n where m = min 1 i n { i }, M = m 1 i n { i }. p; ; t, s) M t r p; ; r, t), n 5.13) The inequlity property in 5.13) follows from the following elementry inequlities in [24, p. 204] which re due to Cuchy. For n rbitrry sequence b = b 1,..., b n ) nd positive sequence c = c 1,..., c n ), we hve min 1 i n { bi c i } n i=1 b i n i=1 c i m 1 i n { bi c i }. 5.14) Equlity holds in both bove inequlities if nd only if the sequences b nd c re proportionl. Uisng Lemm 4.1 nd by stndrd rguments, we obtin the monotonicity of M n p; ; r, s) with respect to vribles r nd s. Theorem 5.7 [30]). The men M n p; ; r, s) of numbers = 1,..., n ) with weights p = p 1,..., p n ) nd two prmeters r nd s is incresing in both r nd s. By mthemticl induction nd inequlities in 5.14), we obtin n inequlity for different nturl indices n of M n p; ; r, s). Theorem 5.8 [30]). For monotonic sequence of positive numbers 0 < 1 2 nd positive weights p = p 1, p 2,... ), if m < n, then Equlity holds if 1 = 2 =. M m p; ; r, s) M n p; ; r, s). 5.15) Using the discrete Tchebysheff s inequlity, the following re obtined. Theorem 5.9 [30]). Let p = p 1,..., p n ) nd q = q 1,..., q n ) be positive weights, = 1,..., n ) sequence of positive numbers. If the sequences p 1 ) q 1,..., pn q n nd re both nonincresing or both nondecresing, then M n p; ; r, s) M n q; ; r, s). 5.16) If one of the sequences of p 1 q 1,..., pn q n ) or is nonincresing nd the other nondecresing, the inequlity 5.16) is reversed. Theorem 5.10 [30]). Let p = p 1,..., p n ) be positive weights, = 1,..., n ) nd b = b 1,..., b n ) two sequences of positive numbers. If the sequences 1 ) b 1,..., n b n nd b re both incresing or both decresing, then M n p; ; r, s) M n p; b; r, s) 5.17) holds for i b i 1, n i 1, nd r, s 0 or r 0 s. The inequlity 5.17) is reversed for i b i 1, n i 1, nd r, s 0 or s 0 r. If one of the sequences of 1 ) b 1,..., n b n or b is nonincresing nd the other nondecresing, then inequlity 5.17) is vlid for i b i 1, n i 1 nd r, s 0 or s 0 r; the inequlity 5.17) reverses for i b i 1, n i 1, nd r, s 0 or r 0 s,.

11 THE EXTENDED MEAN VALUES Generlized bstrcted men vlues. The following definition is n integrl nlogue of the Definition 3 in [24, p. 75]. Definition 5.3. Let p be defined, positive nd integrble function on [, y] for, y R, f rel-vlued nd monotonic function on [α, β]. If g is function vlued on [α, β] nd f g integrble on [, y], the qusi-rithmtic non-symmetricl men of function g is defined by M f g; p;, y) = f 1 y pt)fgt))dt y pt)dt ), 5.18) where f 1 is the inverse function of f. Remrk 5.2. For gt) = t, ft) = t r 1, pt) = 1, the men M f g; p;, y) reduces to the etended logrithmic mens S r, y); for pt) = t r 1, gt) = ft) = t, to the one-prmeter men J r, y); for pt) = f t), gt) = t, to the bstrcted men M f, y); for gt) = t, pt) = t r 1, ft) = t s r, to the etended men vlues Er, s;, y); for ft) = t r, to the weighted men of order r of the function g with weight p on [, y]. If we replce pt) by pt)f r t), ft) by t s r, gt) by ft) in 5.18), then we get the generlized weighted men vlues M p,f r, s;, y). Hence, from M f g; p;, y) we cn deduce lot of the two vrible mens. The following properties follow esily from Lemm 5.1 nd stndrd rguments. Theorem 5.11 [30]). The men M f g; p;, y) hs the following properties α M f g; p;, y) β, M f g; p;, y) = M f g; p; y, ), where α = inf t [,y] gt) nd β = sup t [,y] gt). The function 1 is the inverse function of f) =. Further, we hve Lemm 5.2 [30]). Suppose the rtio f1 ) 1 f1 ) = where 5.19) is monotonic on given intervl. Then f2 f1 ) 1 is the inverse function of f 1. f 1 ) 1 ) 1, 5.20) These hints remind us tht, if replcing 1 s r by f1 ) 1 in Definition 5.2, then we cn obtin Definition 5.4 [30]). Let f 1 nd be rel-vlued functions such tht the rtio f 1 is monotone on the closed intervl [α, β]. If = 1,..., n ) is sequence of rel numbers from [α, β] nd p = p 1,..., p n ) sequence of positive numbers, the generlized bstrcted men vlues of numbers with respect to functions f 1 nd, with weights p, is defined by ) 1 n f1 i=1 M n p; ; f 1, ) = p ) if 1 i ) n i=1 p, 5.21) i i ) where f1 ) 1 is the inverse function of f 1. The integrl nlogue of Definition 5.4 is given by

12 12 F. QI Definition 5.5 [30]). Let p be positive integrble function defined on [, y],, y R, f 1 nd rel-vlued functions nd the rtio f1 monotone on the intervl [α, β]. In ddition, let g be defined on [, y] nd vlued on [α, β], nd f i g integrble on [, y] for i = 1, 2. The generlized bstrcted men vlues of function g with respect to functions f 1 nd nd with weight p is defined s ) 1 y f1 Mp; g; f 1, ;, y) = pt)f ) 1gt))dt y pt)f, 5.22) 2gt))dt where f1 ) 1 is the inverse function of f 1. Remrk 5.3. Set 1 in Definition 5.5, then we cn obtin Definition 5.3 esily. Replcing f by f1, pt) by pt) gt)) in Definition 5.3, we rrive t Definition 5.5 directly. Anlogously, formul 5.21) is equivlent to M f ; p). Definition 5.3 nd Definition 5.5 re equivlent to ech other. Similrly, so re Definition 5.4 nd the qusi-rithmtic non-symmetricl men M f ; p) of numbers = 1,..., n ) with weights p = p 1,..., p n ). From inequlity 5.14), Lemm 5.1, Lemm 5.2 nd stndrd rguments, we hve Theorem 5.12 [30]). The mens M n p; ; f 1, ) nd Mp; g; f 1, ;, y) hve the following properties 1) Under the conditions of Definition 5.4, we hve m M n p; ; f 1, ) M, M n p; ; f 1, ) = M n p; ;, f 1 ), where m = min 1 i n { i }, M = m 1 i n { i }; 2) Under the conditions of Definition 5.5, we hve α Mp; g; f 1, ;, y) β, Mp; g; f 1, ;, y) = Mp; g; f 1, ; y, ), Mp; g; f 1, ;, y) = Mp; g;, f 1 ;, y), where α = inf t [,y] gt) nd β = sup t [,y] gt). By Lemm 5.1 nd stndrd rgument, it follows tht 5.23) 5.24) Theorem ) 5.13 [30]). Suppose p nd g re defined on R. If f 1 g hs constnt sign nd if f1 g is incresing or decresing, respectively), then Mp; g; f 1, ;, y) hve the inverse or sme) monotonicities s f1 with both nd y. The Tchebysheff s integrl inequlity produces the following two theorems. Theorem 5.14 [30]). Suppose g hs constnt sign on [, y]. When gt) increses on [, y], if p1 p 2 is incresing, we hve Mp 1 ; g; f 1, ;, y) Mp 2 ; g; f 1, ;, y); 5.25) if p1 p 2 is decresing, inequlity 5.25) reverses. When gt) decreses on [, y], if p1 p 2 is incresing, then inequlity 5.25) is reversed; if p1 p 2 is decresing, inequlity 5.25) holds. Theorem 5.15 [30]). Suppose g 2 does not chnge its sign on [, y].

13 1) When g1 g 2 re both incresing or both decresing, inequlity holds for f1 g 2 ) THE EXTENDED MEAN VALUES 13 nd ) f1 Mp; g 1 ; f 1, ;, y) Mp; g 2 ; f 1, ;, y) 5.26) being incresing, or reverses for f1 being decresing. ) g 2 or f1 ) g 2 is decresing nd the other 2) When one of the functions g1 incresing, inequlity 5.26) holds for f1 f 1 being incresing. being decresing, or reverses for 5.3. More bsolutely monotonic conve) functions. In [30] nd [31], some more generl bsolutely regulrly, completely) monotonic conve) functions were estblished, which generlize the relted results in [46] restted in Theorem 1.3 of Section 1.3. Theorem 5.16 [31]). Suppose tht fu) is positive nd hs derivtives of ll orders on the intervl [, b]. Define ψt) by f t b) f t ), t 0; ψt) = t 5.27) ln fb) ln f) t = 0. Then ψ n) t) = U nt, fb)) U n t, f)) t n+1, 5.28) U n t, s) = t n+1 ln s) n s t 1, s 5.29) where U n is defined in 1.13). Theorem 5.17 [31]). If fu) 1 nd f u) 0, then the function ψt) defined by 5.27) is bsolutely nd regulrly monotonic on the intervl R. If 0 < fu) 1 nd f u) 0, then ψt) is completely nd regulrly monotonic on R. Moreover, ψt) is bsolutely conve on R. Theorem 5.18 [30]). Suppose F t) = b pu)f t u)du, where t R, pu) 0 is nonnegtive nd continuous function, nd fu) is positive nd continuous function on given intervl [, b]. Then F n) t) = b pu)f t u) [ ln fu) ] n du. 5.30) If fu) 1, then F t) is bsolutely monotone on R; if 0 < fu) < 1, then F t) is completely monotone on R. Moreover, F t) is bsolutely conve on R. 6. Applictions nd relted results The etended men vlues nd their generliztions hve been pplied not only to estblish inequlities of the gmm function nd the incomplete gmm function, to construct new Steffensen pirs, nd to generlize the Hermite-Hdmrd s inequlity, but lso to study qutum nd to generlize the Bernoulli s numbers nd polynomils Appliction to qutum. The concepts of the generlized weighted men vlues M p,f r, s;, y) hve been pplied to study of quntum in [49, 50].

14 14 F. QI 6.2. Generliztions of Bernoulli s numbers nd polynomils. The function gt;, y) defined by 1.12) hs been pplied to generlize the concepts of Bernoulli s numbers nd polynomils. For detils, plese refer to [12, 22, 38] Generliztion of Hermite-Hdmrd s inequlity. Using Tchebycheff s integrl inequlity, the suitble properties of double integrl nd the revised Cuchy s men vlue theorem in integrl form in Lemm 5.1, the following result is proved. Theorem 6.1 [13]). Suppose f) is positive differentible function nd w) 0 n integrble nonnegtive weight on the intervl [, b], if f ) nd f ) w) re integrble nd both incresing or both decresing, then, for ll rel numbers r nd s, we hve M w,f r, s;, b) < E r + 1, s + 1; f), fb) ) ; 6.1) if one of the functions f ) or f ) w) is nondecresing nd the other nonincresing, then inequlity 6.1) reverses. This inequlity 6.1) generlizes Hermite-Hdmrd s inequlity. See [3, 13]. In [27], Hermite-Hdmrd s inequlity ws generlized to the cse of r-conve functions with help of the etended men vlues. In [21], the results obtined in [27] were further generlized to the cse of so-clled g-conve functions Monotonicity results nd inequlities involving gmm functions. It is well-known tht the incomplete gmm function Γz, ) is defined for Re z > 0 by 1.16) nd γz, ) = 0 t z 1 e t dt, 6.2) nd Γz, 0) = Γz) is clled the gmm function, Γ0, ) = E 1 ) the eponentil integrl. In [33], using inequlity 6.1) nd some results on the monotonicities of the generlized weighted men vlues M p,f r, s;, y), it ws verified tht functions [ Γs) Γr) ] 1/s r), [ Γs,) Γr,) ] 1/s r) nd [ γs,) γr,) ] 1/s r) re incresing in r > 0, s > 0 nd > 0. From this, some monotonicity results nd inequlities for the gmm or the incomplete gmm functions re deduced or etended, unified proof of some known results for the gmm function is given. If tking pt) = e t nd ft) = t for t 0, ) in Theorem 6.1, then we hve Theorem 6.2 [33]). For fied > 0, the function sγs,) is decresing in s > 0. s From the monotonicity with the two prmeters r nd s of M p,f r, s;, y) in Theorem 5.3, it follows tht [ 1/s r) Theorem 6.3 [33]). The function Γs) Γr)] is incresing with r > 0 nd s > 0. Corollry [33]). The functions [Γr)] 1/r 1) nd the digmm function ψr) = Γ r) Γr), the logrithmic derivtive of the gmm function Γr), re incresing in r > 0. Hence Γr) is logrithmiclly conve function in the intervl 0, ). Remrk 6.1. In [18] nd [23], mong other things, the following monotonicity results were obtined [Γ1 + k)] 1/k < [Γ2 + k)] 1/k+1), k N;

15 THE EXTENDED MEAN VALUES 15 [ Γ 1 + )] 1 decreses with > 0. Clerly, our Theorem 6.3 nd Corollry generlize nd etend these results for the rnge of the rgument. Corollry The following inequlities hold for s > r > 0 ep [s r)ψs)] > Γs) > ep [s r)ψr)], Γr) 6.3) e cr < Γr + 1) < ep [rψr + 1)], 6.4) where c = is the Euler s constnt. Remrk 6.2. The rtio Γs) Γr) hs been reserched by mny mthemticins. W. Gutschi showed for 0 < s < 1 nd n N in [11] tht n 1 s < Γn + 1) < ep [1 s)ψn + 1)]. 6.5) Γn + s) A strenghened upper bound ws given by T. Erber in [7] s follows Γn + 1) 4n + s)n + 1)1 s < Γn + s) 4n + s + 1) 2, 0 < s < 1, n N. 6.6) J. D. Kečkić nd P. M. Vsić gve in [16] the inequlities below b b 1 1 e b < Γb) Γ) < bb 1/2 1/2 e b, 0 < < b. 6.7) The following closer bounds were proved for 0 < s < 1 nd 1 by D. Kershw in [17]. [ ] [ ep 1 s)ψ + s 1/2 Γ + 1) ) < Γ + s) < ep 1 s)ψ + s + 1 )], 6.8) 2 + s ) 1 s Γ + 1) < 2 Γ + s) < [ 12 + s 1 4) 1/2 ] 1 s. 6.9) It is esy to see tht inequlities in 6.3) of Corollry etend the rnge of rguments of bove inequlities 6.5) 6.9) but 6.7). As consequences of Theorem 5.2 nd Theorem 5.3, we hve Theorem 6.4 [33]). For s > r > 0 nd > 0, the functions ] 1/s r) increse with either or r nd s. Therefore, γs,) s 1 [ Γs,) Γr,) Γs,) s 1 increses with s > 0, respectively. [ γs,) γr,) ] 1/s r) nd decreses nd Corollry The incomplete gmm functions γr, ) nd Γr, ) re logrithmiclly conve with respect to r > 0 for fied > 0. The function is incresing in r > 0 nd > 0. Therefore, the functions Γs+θ) γs+θ,) γr+θ,) re incresing with θ for fied s > r > 0 nd > 0. [ ] 1/r Γr,) E 1) Γr+θ), Γs+θ,) Γr+θ,) nd Remrk 6.3. In the lst week of November 2001, N. Elezović reminded me of his joint pper [5] with C. Giordn nd J. Pecrić. In their pper [5], mong others, the [ 1/t s) conveity with respect to vrible of the function Γ+t) Γ+s)] for t s < 1

16 16 F. QI is verified, the best lower bound for 6.8) nd the best upper bound for 6.9) re obtined, some different pproch from Gutschi s in [11] is given, severl new simple inequlities for digmm function re lso proved. The gmm nd incomplete gmm functions nd relted functions hve been investigted using different pproches, for emples, see [1, 4, 37, 40, 41, 43] Estblishment of Steffensen pirs. Let f nd g be integrble functions on [, b] such tht f is decresing nd 0 g) 1 for [, b]. Then b b λ f)d b f)g)d +λ f)d, 6.10) where λ = b g)d. The inequlity 6.10) is clled Steffensen s inequlity. In [8], discrete nlogue of the inequlity 6.10) ws proved: Let { i } n i=1 be decresing finite sequence of nonnegtive rel numbers, {y i } n i=1 be finite sequence of rel numbers such tht 0 y i 1 for 1 i n. Let k 1, k 2 {1, 2,, n} be such tht k 2 n i=1 y i k 1. Then n n k 1 i i y i i. 6.11) i=n k 2+1 i=1 As direct consequence of inequlity 6.11), we hve: Let { i } n i=1 be nonnegtive rel numbers such tht n i=1 i A nd n i=1 2 i B2, where A nd B re positive rel numbers. Let k {1, 2,, n} be such tht k A B. Then there re k numbers mong 1, 2,..., n whose sum is bigger thn or equls to B. The so-clled Steffensen pir ws defined by H. Guchmn in [10] s follows. Definition 6.1. Let ϕ : [c, ) [0, ) nd τ : 0, ) 0, ) be two strictly incresing functions, c 0, let { i } n i=1 be finite sequence of rel numbers such tht i c for 1 i n, A nd B be positive rel numbers, nd n i=1 i A, n i=1 ϕ i) ϕb). If, for ny k {1, 2,, n} such tht k τ ) A B, there re k numbers mong 1,..., n whose sum is not less thn B, then we cll ϕ, τ) Steffensen pir on [c, ). The following Steffensen pirs were found by H. Guchmn in [10]. α, 1/α 1)), α 2, [0, ); 6.12) ep α 1), 1 + ln ) 1/α), α 1, [1, ). 6.13) Let nd b be rel numbers stisfying b > > 1 nd b e. Define 1+ln b 1+ln if > 1, ϕ) = ln ln b ln if = 1, i=1 6.14) τ) = 1/ ln b. 6.15) Then it ws verified by H. Guchmn in [10] tht ϕ, τ) is Steffensen pir on [1, ) using some results nd techniques in [46]. With help of properties of the etended men vlues Er, s;, y) nd the generlized weighted men vlues M p,f r, s;, y), some new Steffensen pirs were estblished in [36, 39].

17 THE EXTENDED MEAN VALUES 17 Using the integrl epression 1.14) of function b, mthemticl induction nd nlytic techniques, we hve Theorem 6.5 [36]). If nd b re rel numbers stisfying b > > 1 or b > 1 > 1, nd b e, then ) b t ln 1 dt, 2/ ln b) 6.16) is Steffensen pir on [1, ). If nd b re rel numbers stisfying b > > 1 nd b e, then ) b ln t) n t ln 1 dt, n+2 n+1 ln b)n+1 ln ) n+1 ln b) n+2 ln ) n ) re Steffensen pirs on [1, ) for ny positive integer n. In [39], considering the function b pu)f t u)du nd its properties, we further obtin much generl Steffensen pirs s follows. Theorem 6.6 [39]). Let, b R, let p 0 be nonnegtive nd integrble function nd f positive nd integrble function on the intervl [, b]. 1) If inequlity holds, then b b pu)du b pu)[fu)] ln du, is Steffensen pir on [1, ). 2) If fu) 1 nd inequlity 6.18) holds, then b pu)[fu)] ln [ln fu)] n du, pu) ln fu)du 6.18) b ) pu)du b pu) ln fu)du b pu)[ln fu)] n du b pu)[ln fu)] n+1 du re Steffensen pirs on [1, ) for ny positive integer n. ) 6.19) 6.20) Acknowledgements. This pper ws completed during the uthor s visit to the RGMIA between November 1, 2001 nd Jnury 31, 2002, s Visiting Professor with grnts from the Victori University of Technology nd Jiozuo Institute of Technology. References [1] G. Allsi, C. Giordno, nd J. Pečrić, Hdmrd-type inequlities for 2r)-conve functions with pplictions, Atti Accd. Sci. Torino Cl. Sci. Fis Mt Ntur ), [2] E. F. Beckenbch nd R. Bellmn, Inequlities, Springer, Berlin, [3] S. S. Drgomir nd C. E. M. Perce, Selected Topics on Hermite-Hdmrd Type Inequlities nd Applictions, RGMIA Monogrphs, Avilble online t monogrphs/hermite_hdmrd.html. [4] Á. Elbert nd A. Lforgi, An inequlity for the product of two integrls relting to the incomplete gmm function, J. Inequl. Appl ), [5] N. Elezović, C. Giordno nd J. Pecrić, The best bounds in Gutschi s inequlity, Mth. Inequl. Appl ), no. 2, [6] N. Elezović nd J. Pečrić, A note on Schur-conve functions, Rocky Mountin J. Mth ), no. 3,

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