Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 305623, 12 pages doi:10.1155/2008/305623 Research Article On Logarithmic Convexity for Power Sums and Related Results II J. Pe ˇ cari ´ c 1, 2 andAtiqurRehman 1 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan 2 Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia Correspondence should be addressed to Atiq ur Rehman, mathcity@gmail.com Received 14 October 2008; Accepted 4 December 2008 Recommended by Wing-Sum Cheung In the paper “On logarithmic convexity for power sums and related results” 2008, we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums. Also we give integral analogs of new means. Copyright q 2008 J. Pe ˇ cari ´ c and A. ur Rehman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries Let x be positive n-tuples. The well-known inequality for power sums of order s and r,for s>r>0 see 1, page 164, states that  n  i1 x s i  1/s <  n  i1 x r i  1/r . 1.1 Moreover, if p p 1 , ,p n  is a positive n-tuples such that p i ≥ 1 i  1, ,n, then for s>r>0 see 1, page 165, we have  n  i1 p i x s i  1/s <  n  i1 p i x r i  1/r . 1.2 In 2, we defined the following function: Δ t Δ t x; p ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 t − 1  n  i1 p i x i  t − n  i1 p i x t i  ,t /  1, n  i1 p i x i log n  i1 p i x i − n  i1 p i x i log x i ,t 1. 1.3 2 Journal of Inequalities and Applications We introduced the Cauchy means involving power sums. Namely, the following results were obtained in 2. For r<s<t,where r, s, t ∈ R  , we have  Δ s  t−r ≤  Δ r  t−s  Δ t  s−r , 1.4 such that, x i ∈ 0,ai  1, ,n and n  i1 p i x i ≥ x j , for j  1, ,n, n  i1 p i x i ∈ 0,a. 1.5 We defined the following means. Definition 1.1. Let x and p be two nonnegative n-tuples n ≥ 2 such that p i ≥ 1 i  1, ,n. Then for t, r, s ∈ R  , A s t,r x; p  r − s t − s   n i1 p i x s i  t/s −  n i1 p i x t i   n i1 p i x s i  r/s −  n i1 p i x r i  1/t−r ,t /  r, r /  s, t /  s, A s s,r x; pA s r,s x; p  r − s s   n i1 p i x s i  log  n i1 p i x s i − s  n i1 p i x s i log x i   n i1 p i x s i  r/s −  n i1 p i x r i  1/s−r ,s /  r, A s r,r x; pexp  1 s − r    n i1 p i x s i  r/s log  n i1 p i x s i − s  n i1 p i x r i log x i s   n i1 p i x s i  r/s −  n i1 p i x r i   ,s /  r, A s s,s x; pexp    n i1 p i x s i  log  n i1 p i x s i  2 − s 2  n i1 p i x s i  log x i  2 2s   n i1 p i x s i  log   n i1 p i x s i  − s  n i1 p i x s i log x i   . 1.6 In this paper, we introduce new Cauchy means of convex type in connection with Power sums. For means, we shall use the following result 1, page 154. Theorem 1.2. Let x and p be two nonnegative n-tuples such that condition 1.5 is valid. If f is a convex function on 0,a,then f  n  i1 p i x i  ≥ n  i1 p i f  x i    1 − n  i1 p i  f0. 1.7 Remark 1.3. In Theorem 1.2,iff is strictly convex, then 1.7 is strict unless x 1  ···  x n and  n i1 p i  1. J. Pe ˇ cari ´ c and A. ur Rehman 3 2. Discrete result Lemma 2.1. Let ϕ t x ⎧ ⎪ ⎨ ⎪ ⎩ x t tt − 1 ,t /  1, x log x, t  1, 2.1 where t ∈ R  .Thenϕ t x is strictly convex for x>0. Here,weusethenotation0log0: 0. Proof. Since ϕ  t xx t−2 > 0forx>0, therefore ϕ t x is strictly convex for x>0. Lemma 2.2 see 3. A positive function f is log convex in Jensen sense on an open interval I, that is, for each s, t ∈ I fsft ≥ f 2  s  t 2  , 2.2 if and only if the relation u 2 fs2uwf  s  t 2   w 2 ft ≥ 0, 2.3 holds for each real u, w and s, t ∈ I. The following lemma is equivalent to definition of convex function 1, page 2. Lemma 2.3. If f is continuous and convex for all x 1 , x 2 , x 3 of an open interval I for which x 1 <x 2 < x 3 ,then  x 3 − x 2  f  x 1    x 1 − x 3  f  x 2    x 2 − x 1  f  x 3  ≥ 0. 2.4 Lemma 2.4. Let f be log-convex function and if, x 1 ≤ y 1 ,x 2 ≤ y 2 ,x 1 /  x 2 ,y 1 /  y 2 , then the following inequality is valid:  f  x 2  f  x 1   1/x 2 −x 1  ≤  f  y 2  f  y 1   1/y 2 −y 1  . 2.5 By using the above lemmas and Theorem 1.2,asin2, we can prove the following results. Theorem 2.5. Let x and p be two positive n-tuples and let Δ t  Δ t (x; p Δ t t , 2.6 4 Journal of Inequalities and Applications such that condition 1.5 is satisfied and all x i ’s are not equal. Then Δ t is log-convex. Also for r<s<t where r, s, t ∈ R  , we have  Δ s  t−r ≤  Δ r  t−s  Δ t  s−r . 2.7 Moreover, we can use 2.7 to obtain new means of Cauchy type involving power sums. Let us introduce the following means. Definition 2.6. Let x and p be two nonnegative n-tuples such that p i ≥ 1 i  1, ,n, then for t, r, s ∈ R  , B s t,r x; p  rr − s tt − s   n i1 p i x s i  t/s −  n i1 p i x t i   n i1 p i x s i  r/s −  n i1 p i x r i  1/t−r ,t /  r, r /  s, t /  s, B s s,r x; pB s r,s x; p  rr − s s 2   n i1 p i x s i  log  n i1 p i x s i − s  n i1 p i x s i log x i   n i1 p i x s i  r/s −  n i1 p i x r i  1/s−r ,s /  r, B s r,r x; pexp  − 2r − s rr − s    n i1 p i x s i  r/s log  n i1 p i x s i − s  n i1 p i x r i log x i s   n i1 p i x s i  r/s −  n i1 p i x r i   ,s /  r, B s s,s x; pexp  − 1 s    n i1 p i x s i  log  n i1 p i x s i  2 − s 2  n i1 p i x s i  log x i  2 2s   n i1 p i x s i  log  n i1 p i x s i − s  n i1 p i x s i log x i   . 2.8 Remark 2.7. Let us note that B s s,r x; pB s r,s x; plim t → s B s t,r x; plim t → s B s r,t x; p, B s r,r x; plim t → r B s t,r x; p and B s s,s x; plim r → s B s r,r x; p. Theorem 2.8. Let Θ s t  ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 tt − s  n  i1 p i x s i  t/s − n  i1 p i x t i  ,t /  s, 1 s 2  n  i1 p i x s i  log  n  i1 p i x s i  − s n  i1 p i x s i log x i  ,t s. 2.9 then for t, r, u ∈ R  and t<r<u, we have  Θ s r  u−t ≤  Θ s t  u−r  Θ s u  r−t . 2.10 Theorem 2.9. Let r, t, u, v ∈ R  , such that t ≤ v, r ≤ u. Then one has B s t,r (x; p ≤ B s v,u (x; p. 2.11 J. Pe ˇ cari ´ c and A. ur Rehman 5 Remark 2.10. From 2.7, we have  Δ s s  t−r ≤  Δ r r  t−s  Δ t t  s−r ⇒  Δ s  t−r ≤ s t−r r t−s t s−r  Δ r  t−s  Δ t  s−r . 2.12 Since log x is concave, therefore f or r<s<t, we have t − s log r r − t log s s − r log t<0 ⇒ s t−r r t−s t s−r > 1. 2.13 This implies that 1.4, which we derived in 2, is better than 2.7. Also note that B s t,r x; p  r t  1/t−r A s t,r x; p, B s r,s x; pB s s,r x; p  r s  1/s−r A s s,r x; p  r s  1/s−r A s r,s x; p, B s r,r x; pexp  − 1 r  A s r,r x; p, B s s,s x; pexp  − 1 s  A s s,s x; p. 2.14 Let us note that there are not integral analogs of results from 2. Moreover, in Section 3 we will show that previous results have their integral analogs. 3. Integral results The following theorem is very useful for further result 1, page 159. Theorem 3.1. Let t 0 ∈ a, b be fixed, h be continuous and monotonic with ht 0 0, g be a function of bounded variation and Gt :  t a dgx, Gt :  b t dgx. 3.1 a If 0 ≤ Gt ≤ 1 for a ≤ t ≤ t 0 , 0 ≤ Gt ≤ 1 for t 0 ≤ t ≤ b, 3.2 then for every convex function f : I → R such that hx ∈ I for all x ∈ a, b,  b a f  ht  dgt ≥ f   b a htdgt     b a dgt − 1  f0. 3.3 6 Journal of Inequalities and Applications b If  b a htdgt ∈ I and either there exists an s ≤ t 0 such that Gt ≤ 0 for t < s, Gt ≥ 1 for s ≤ t ≤ t 0 , Gt ≤ 0 for t>t 0 , 3.4 or there exists an s ≥ t 0 such that Gt ≤ 0 for t<t 0 , Gt ≥ 1 for t 0 <t<s, Gt ≤ 0 for t ≥ s, 3.5 then for every convex function f : I → R such that hx ∈ I for all x ∈ a, b, the reverse of the inequality in 3.3 holds. To define the new means of Cauchy involving integrals, we define the following function. Definition 3.2. Let t 0 ∈ a, b be fixed, h be continuous and monotonic with ht 0 0, g be a function of bounded variation. Choose g such that function Λ t is positive valued, where Λ t is defined as follows: Λ t Λ t a, b, h, g  b a ϕ t  hx  dgx − ϕ t   b a hxdgx  . 3.6 Theorem 3.3. Let Λ t , defined as above, satisfy condition 3.2.ThenΛ t is log-convex. Also for r< s<t,wherer, s, t ∈ R  , one has  Λ s  t−r ≤  Λ r  t−s  Λ t  s−r . 3.7 Proof. Let fxu 2 ϕ s x2uwϕ r xw 2 ϕ t x, where r s  t/2andu, w ∈ R, f  xu 2 x s−2  2uwx r−2  w 2 x t−2   ux s−2/2  wx t−2/2  2 ≥ 0. 3.8 This implies that fx is convex. By Theorem 3.1, we have,  b a f  ht  dgt − f   b a htdgt  −   b a dgt − 1  f0 ≥ 0 ⇒ u 2   b a ϕ s hxdgx − ϕ s   b a hxdgx   2uw   b a ϕ r hxdgx − ϕ r   b a hxdgx   2w 2   b a ϕ t hxdgx − ϕ t   b a hxdgx  ≥ 0 ⇒ u 2 Λ s  2uwΛ r  w 2 Λ t ≥ 0. 3.9 Now, by Lemma 2.2, we have Λ t is log-convex in Jensen sense. J. Pe ˇ cari ´ c and A. ur Rehman 7 Since lim t → 1 Λ t Λ 1 ,thisimpliesthatΛ t is continuous for all t ∈ R  , therefore it is a log-convex 1, page 6. Since Λ t is log-convex, that is, log Λ t is convex, therefore by Lemma 2.3 for r<s<t and taking f  log Λ, we have t − s log Λ r r − t log Λ s s − r log Λ t ≥ 0, 3.10 which is equivalent to 3.7. Theorem 3.4. Let  Λ t  −Λ t such that condition 3.4 or 3.5 is satisfied. Then  Λ t is log-convex. Also for r<s<t,wherer, s, t ∈ R  , one has   Λ s  t−r ≤   Λ r  t−s   Λ t  s−r . 3.11 Definition 3.5. Let t 0 ∈ a, b be fixed, h be continuous and monotonic with ht 0 0, g be a function of bounded variation. Then for t, r, s ∈ R  , one defines F s t,r a, b, h, g  ⎧ ⎨ ⎩ rr − s tt − s  b a h t xdgx −   b a hxdgx  t/s  b a h r xdgx −   b a hxdgx  r/s ⎫ ⎬ ⎭ 1/t−r ,t /  r, r /  s, t /  s, F s s,r a, b, h, g  F s r,s a, b, h, g  ⎧ ⎨ ⎩ rr − s s 2 s  b a h s x log hxdgx −   b a h s xdgx  log  b a h s xdgx  b a h r xdgx −   b a h s xdgx  r/s ⎫ ⎬ ⎭ 1/s−r ,s /  r, F s r,r a, b, h, g  exp  − 2r − s rr − s  s  b a h r x log hxdgx−   b a h s xdgx  r/s log  b a h s xdgx s   b a h r xdgx−   b a h s xdgx  r/s   ,s /  r, F s s,s a, b, h, g  exp  − 1 s  s 2  b a h s x  log hx  2 dgx −   b a h s xdgx  log  b a h s xdgx  2 2s  s  b a h s x log hxdgx −   b a h s xdgx  log  b a h s xdgx   . 3.12 Remark 3.6. Let us note that F s s,r a, b, h, gF s r,s a, b, h, glim t→s F s t,r a, b, h, g lim t→s F s r,t a, b, h, g, F s r,r a, b, h, glim t→r F s t,r a, b, h, g and F s s,s a, b, h, glim r→s F s r,r a, b, h, g. 8 Journal of Inequalities and Applications Theorem 3.7. Let r, t, u, v ∈ R  , such that t ≤ v, r ≤ u.Then F s t,r a, b, h, g ≤ F s v,u a, b, h, g. 3.13 Proof. Let Λ t Λ t a, b, h, g  ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 tt − 1   b a h t xdgx −   b a hxdgx  t  ,t /  1,  b a hx log hxdgx −  b a hxdgx log  b a hxdgx,t 1. 3.14 Now, taking x 1  r, x 2  t, y 1  u, y 2  v, where r, t, u, v /  1, and ftΛ t in Lemma 2.4,we have ⎛ ⎝ rr − 1 tt − 1  b a h t xdgx −   b a hxdgx  t  b a h r xdgx −   b a hxdgx  r ⎞ ⎠ 1/t−r ≤ ⎛ ⎝ uu − 1 vv − 1  b a h v xdgx −   b a hxdgx  v  b a h u xdgx −   b a hxdgx  u ⎞ ⎠ 1/v−u . 3.15 Since s>0, by substituting h  h s , t  t/s, r  r/s, u  u/s, and v  v/s, where r, t, v, u /  s, in above inequality, we get ⎛ ⎝ rr − s tt − s  b a h t xdgx −   b a h s xdgx  t/s  b a h r xdgx −   b a h s xdgx  r/s ⎞ ⎠ s/t−r ≤ ⎛ ⎝ uu − s vv − s  b a h v xdgx −   b a h s xdgx  v/s  b a h u xdgx −   b a h s xdgx  u/s ⎞ ⎠ s/v−u . 3.16 By raising power 1/s, we get an inequality 3.13 for r, t, v, u /  s. From Remark 3.6 ,weget3.13 is also valid for r  s or t  s or r  t or t  r  s. Lemma 3.8. Let f ∈ C 2 I such that m ≤ f  x ≤ M. 3.17 J. Pe ˇ cari ´ c and A. ur Rehman 9 Consider the functions φ 1 , φ 2 defined as φ 1 x Mx 2 2 − fx, φ 2 xfx − mx 2 2 . 3.18 Then φ i x for i  1, 2 are convex. Proof. We have that φ  1 xM − f  x ≥ 0, φ  2 xf  x − m ≥ 0, 3.19 that is, φ i for i  1, 2 are convex. Theorem 3.9. Let t 0 ∈ a, b be fixed, h be continuous and monotonic with ht 0 0, g be a function of bounded variation, and f ∈ C 2 I such that condition 3.2 is satisfied. Then there exists ξ ∈ I such that  b a f  hx  dgx − f   b a hxdgx  −   b a dgx − 1   f  ξ 2   b a h 2 xdgx −   b a hxdgx  2  . 3.20 Proof. In Theorem 3.1, setting f  φ 1 and f  φ 2 , respectively, as defined in Lemma 3.8,we get the following inequalities:  b a f  hx  dgx − f   b a hxdgx  −   b a dgx − 1  ≤ M 2   b a h 2 xdgx −   b a hxdgx  2  , 3.21  b a f  hx  dgx − f   b a hxdgx  −   b a dgx − 1  ≥ m 2   b a h 2 xdgx −   b a hxdgx  2  . 3.22 Now, by combining both inequalities, we get m ≤ 2   b a f  hx  dgx − f   b a hxdgx  −   b a dgx − 1  f0   b a h 2 xdgx −   b a hxdgx  2 ≤ M. 3.23 10 Journal of Inequalities and Applications So by condition 3.17, t here exists ξ ∈ I such that 2   b a f  hx  dgx − f   b a hxdgx  −   b a dgx − 1  f0   b a h 2 xdgx −   b a hxdgx  2  f  ξ, 3.24 and 3.24 implies 3.20. Moreover, 3.21 is valid if f  is bounded from above and again we have 3.20 is valid. Of course 3.20 is obvious if f  is not bounded from above and below as well. Theorem 3.10. Let t 0 ∈ a, b be fixed, h be continuous and monotonic with ht 0 0, g be a function of bounded variation, and f 1 ,f 2 ∈ C 2 I such that condition 3.2  is satisfied. Then there exists ξ ∈ I such that the following equality is true:  b a f 1  hx  dgx − f 1   b a hxdgx  −   b a dgx − 1  f 1 0  b a f 2  hx  dgx − f 2   b a hxdgx  −   b a dgx − 1  f 2 0  f  1 ξ f  2 ξ , 3.25 provided that denominators are nonzero. Proof. Let a function k ∈ C 2 I be defined as k  c 1 f 1 − c 2 f 2 , 3.26 where c 1 and c 2 are defined as c 1   b a f 2  hx  dgx − f 2   b a hxdgx  −   b a dgx − 1  f 2 0, c 2   b a f 1  hx  dgx − f 1   b a hxdgx  −   b a dgx − 1  f 1 0. 3.27 Then, using Theorem 3.9 with f  k, we have 0   c 1 f  1 ξ − c 2 f  2 ξ    b a h 2 xdgx −   b a hxdgx  2  . 3.28 Since  b a h 2 xdgx −   b a hxdgx  2 /  0, 3.29 [...]... Education and Sports under the research Grant 117-1170889-0888 12 Journal of Inequalities and Applications References 1 J Peˇ ari´ , F Proschan, and Y L Tong, Convex Functions, Partial Orderings, and Statistical Applications, c c vol 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992 2 J Peˇ ari´ and A U Rehman, On logarithmic convexity for power sums and related results, ”... be continuous and monotonic with h t0 function of bounded variation, and let t, r, s ∈ R Then s Ft,r a, b, h, g η Proof If t, r, and s are pairwise distinct, then we put α x xt , β x 3.32 to get 3.34 For other cases, we can consider limit as in Remark 3.6 0, g be a 3.34 xr and γ x xs in Acknowledgments This research was partially funded by Higher Education Commission, Pakistan The research of the first... ari´ and A U Rehman, On logarithmic convexity for power sums and related results, ” Journal c c of Inequalities and Applications, vol 2008, Article ID 389410, 9 pages, 2008 3 S Simic, On logarithmic convexity for differences of power means,” Journal of Inequalities and Applications, vol 2007, Article ID 37359, 8 pages, 2007 ...J Peˇ ari´ and A ur Rehman c c 11 therefore, 3.28 gives c2 c1 f1 ξ f2 ξ 3.30 After putting values, we get 3.25 Let α be a strictly monotone continuous function, we defined Tα h, g as follows integral version of quasiarithmetic sum 2 : α−1 Tα h, g b α h x dg x 3.31 a Theorem 3.11 Let α, β, γ ∈ C2 a, b be strictly monotonic continuous functions Then there exists η in the image... nonzero Proof If we choose the functions f1 and f2 so that f1 Substituting these in 3.25 , α Tα h, g − α Tγ h, g − β Tγ h, g b dg a b dg a − β Tβ h, g α◦γ −1 , f2 − β◦γ −1 , and h x → γ h x x − 1 α ◦ γ −1 0 x − 1 β ◦ γ −1 0 α γ −1 ξ γ γ −1 ξ β γ −1 ξ γ γ −1 ξ Then by setting γ −1 ξ − α γ −1 ξ γ γ −1 ξ − β γ −1 ξ γ γ −1 ξ 3.33 η, we get 3.32 Corollary 3.12 Let t0 ∈ a, b be fixed, h be continuous and . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 305623, 12 pages doi:10.1155/2008/305623 Research Article On Logarithmic Convexity for Power Sums and Related Results. Pe ˇ cari ´ c and A. U. Rehman, On logarithmic convexity for power sums and related results, ” Journal of Inequalities and Applications, vol. 2008, Article ID 389410, 9 pages, 2008. 3 S. Simic, On logarithmic. for power sums and related results 2008, we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums.