Geometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions

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1 Geometriclly Convex Function nd Estimtion of Reminder Terms in Tylor Series Expnsion of some Functions Xioming Zhng Ningguo Zheng December Abstrct In this pper two integrl inequlities of geometriclly convex functions re proved. For their ppliction estimtion formuls of reminder terms in Tylor series expnsion of e x sinx nd cosx re given. Key words: inequlity geometriclly convex function definite integrl reminder term. 1 Introduction nd Definition The pper first proves two integrl inequlity of geometriclly concve functions then obtins the estimtion formuls of reminder terms in Tylor series expnsion of e x sint nd cost where x + nd t π 2. In [1] [2] [3] the uthors obtined some reltive definitions of geometriclly convex function. DEFINITION 1. Let f : I + + be continuous function. then f is clled geometriclly convex function on I if there exists n 2 such tht one of the following inequlities holds for ny x 1 x 2 x n I nd α β λ 1 λ 2 λ n > with f x 1 x 2 f x 1 f x 2. 1 f f n x α 1 x β 2 f α x 1 f β x 2. 2 n i=1 x i n n i=1 f x i. 3 Mthemtics Subject Clssifictions: 26D15 26D1. Zhejing Brodcst nd TV University Hining College Zhejing 3144 the People s Republic of Chin. emil: zjzxm79@tom.com Huzhou Brodcst nd TV University Huzhou Zhejing 313 the People s Republic of Chin 1

2 n f i=1 xλi i n f λi x i. 4 i=1 And f is clled geometriclly concve function on I if one of inequlities 1-4 is inverse. LEMMA 1. [5] Let b + f : b + be twice differentible. Then f is geometriclly convex function if nd only if [ x f x f x f x 2] + f x f x 5 hold for every x b. Menwhile f is geometriclly concve function if nd only if inequlity 5 is inverse. LEMMA 2. [2][5] Let < < b f : [ b] + be geometriclly convexconcve function nd g x = ln f e x x [ln ln b]. Then g is convexconcve function. For convexconcve functions unilterl Derivtive exists. Lemm 2 tells us tht unilterl Derivtive of geometriclly convexconcve functions exist. LEMMA 3. [8] Let < b function f be continuous on [ b] be geometriclly concve function on b] nd g x = f t dt x b]. Then g is geometriclly concve function on b]. 2 Two integrl inequlities THEOREM 1. Let < < b f : [ b] + be geometriclly convex function. If f + f + f x dx nd if f b + bf 1 b f x dx f 2 f + f + f + f f b 2 f b + bf b b bf b fb b b f + 1+ f bf 1+ b fb 1+ f + f bf b fb. 7 The equlity occurs only when f is power function or constnt function. Menwhile inequlity 67 re inverse if f is geometriclly concve function. Proof. Let f be geometriclly convex function. ln f c ln f ln f e ln c ln f ln e lim = lim c + ln c ln c + ln c ln = ln f e t + = et f + e t t=ln f e t = f + 1. t=ln f 2

3 ln fc ln f Then we cn choose c b such tht 1. Suppose x [c b] ln x ln c t = ln x ln. Hence t < 1 x t = x c c = t x 1 t nd f c = f t x 1 t. With respect to the definition 1 we hve Hence c f c f t f x 1 t 8 f c 1 1 t f t 1 t f x ln x ln ln x ln c f c f f x. f x dx = f c ln f Let ln x = u u = e x we hve c ln f x dx f c f = f c ln f ln x ln ln x ln c f c f dx c ln c b c ln c ln b ln c ln b ln c = f c ln b f c ln b f 1 + ln b = b f c ln b ln b exp = f c ln x f c f ln c u ln x dx. f c f d e u [ ] 1 e f c f 1 u du f ln c ln c c f c f ln fc ln f b f 1 + b = f c 1 + ln b ln c Let c + in inequlity 9 we get b exp f x dx f ln fc ln f fc f ln b ln c c ln fc ln f c f c ln c ln f c ln f c ln fc ln f ln b ln f + f 1 + f + f = f 2 b f + b f f + f + u 3

4 f 2 = f + f + f + f b f + 1+ f 1+ f + f. Equlity 6 occurs only when equlity 8 occurs f is power function or constnt function. Anlogously we cn proof inequlity 7. The proof of Theorem 1 is completed. THEOREM 2. Let < < b f : [ b] + be geometriclly convex function. Then { b bfb f f x dx lnbfb lnf ln b f bf b f ln b f = bf b. 1 The equlity occurs only when f is power function or constnt function. Menwhile inequlity 1 is inverse if f is geometriclly concve function. Proof. Let f is geometriclly convex function. x = 1 α b α. Then f x dx = 1 For f x dx Let α = log b f 1 α b α 1 α b α ln b dα. With respect to the definition 1 if f bf b we hve = = f ln b f x dx 1 1 f 1 α f α b 1 α b α ln b dα α bf b dα = f f ln bf b f ln b [ bf b f 1 If f = bf b f ln b ln bfb f ] = α bf b 1 f bf b f ln bf b ln f ln b. x is obvious. f x dx f ln b 3 Estimtion Formul of e x THEOREM 3. Let x > n N n 1 nd T n x = e x 1 + x x2 2! n 1 x n n!. 1If < x < n+2 n+1. Then 2 n + 1 n + x + n x 2 4x x n+1 n + 1! T n x n + 2 n x x n+1 n + 1!. 11 4

5 2If x n+2 n+1. Then 2 n + 1 n + x + n x 2 n + 1! T 2 n + 1 n x 4x n x + x + n n + 1!. 4x 12 x n+1 Proof. If n is n odd number let n = 2m 1 m N m 1. If n is n even number let n = 2m m N m 1. According to Tylor series expnsion of e x we hve T 2m 1 x = e x 1 + x x2 2! + + x2m 1 2m 1! T 2m x = e x 1 + x x2 2! + x2m 2m! T 2m x = e x + 1 x + x2 2! + + x2m 2m!. Suppose f x = e x x + then f is geometriclly concve function ccording to Lemm 1. Further since e t dt = 1 e x 1 e t dt = e x 1 + x e t 1 + x dt = e x + 1 x + x2 2. T 2m 1 nd T 2m re both geometriclly concve functions ccording to Lemm 3. Suppose < < x ccording to Theorem 1 we get where η = xt 2m 1 x T 2m 1x T 2m 1 xdx T 2m 1 x 1 + η x η x 1+η 1+η 13 = x e x +1 x++ x2m 2 2m 2! T 2m 1x >. Let + in 13 e x + 1 x + x2 2! + + x2m 2m! T 2m 1 x + x2m 2m! T 2m 1 x + x2m 2m! xt 2m 1 x 1 + xt 2m 1 x T 2m 1x xt2m 1 2 x T 2m 1 x + x e x + 1 x + + x2m 2 2m 2! T 2m 1 x + x x 1 T2m 1 2 x+ x2m 2m 1! + x2m 2m! x2m+1 2m! xt2m 1 2 x T 2m 1 x + x2m 1 2m 1! x 4m x n+1 T 2m 1 x+ 2m! 2m 1! xt 2m 1 2 x 5

6 T 2 2m 1 x + 2m 1 + x x2m T 2m 1 x 2m! T 2m 1 x 2m 1+xx 2m 2m! + = x 4m 2m! 2m 1! 2m 1+x 2 x 4m 2m! 2 + 4x 4m 2m!2m 1! 2 2m 1 + x + 2m 1 + x 2 + 8m 2 x 2m 2m! 4m x 2m = 2m 1 + x 2 2m! + 8m + 2m 1 + x 4m x 2m = 2m x 2 2m!. 14 4x + 2m 1 + x Since T 2m lso is geometriclly concve function by Theorem 1 nlogously we hve 4m + 2 x 2m+1 T 2m x 2m + x + 2m x 2 2m + 1!. 15 4x Let < < x ccording to Theorem 2 we get T 2m 1 t dt T 2m 1 t dt xt 2m 1 x T 2m 1 Let + we note the L Hospitl Rule xt 2m 1 x T 2m 1 ln xt 2m 1 x ln T 2m 1 ln x T 2m 1 t dt xt 2m 1 x lim + e x + 1 x + x2 2! + + x2m 2m! T 2m 1 x + x2m 2m! T 2m 1 x + x2m 2m! ln x ln ln xt 2m 1 x ln T 2m 1. 1/ T2m 1+T 2m 1 T 2m 1 xt 2m 1 x 1 + lim + xt 2m 1 x 2 + lim + T 2m 1 T 2m 1 xt 2m 1 x 2m + lim + e 1 e T 2m 1 x + x2m 2m! xt 2m 1 x 2m + 1 2m + 1 x 2m 2m x T 2m 1 x 2m! 6 T 2m 1 T 2m 1

7 Also for T 2m x similrly we cn get 2m + 1 T 2m 1 x 2m x x 2m 2m!. 16 T 2m x 2m + 2 2m x x 2m+1 2m + 1!. 17 We omit the detils. Since T 2m 1 x is geometriclly concve function ccording to Lemm 1 we hve x T 2m 1 x T 2m 1 x T 2m 1 x 2 + T 2m 1 x T 2m 1 x. 18 Menwhile T 2m 1 x = T 2m 1 x + x2m 1 2m 1! So T 2m 1 x = T 2m 1 x + x2m 2 2m 2! x2m 1 2m 1!. T2m 1 2 x + 2mx2m 1 + x 2m x 4m 1 T 2m 1 x 2m 1! 2m 1! 2 T2m 1 2 x 2mx2m 1 + x 2n T 2m 1 x + 2m 1! x + 2m 2 4x T 2m 1 2m + x 2 x 4m 1 2m 1! 2 x 2m 1 2m 1! = 2m + x + 2x 2m + x 2 4x x 2m 1 2m 1! = 4m 2m + x + 2m + x 2 4x Also for T 2m x fter some similrly clculus we cn get x 2m 2m!. 19 4m + 2 x 2m+1 T 2m x 2m x + 2m x 2 2m + 1! 2 4x If < x 2n+1 2n we cn find inequlity 16 is stronger thn inequlity 19. If x 2n+1 2n+2 2n inequlity 19 is stronger thn inequlity 16. If < x 2n+1 we cn find inequlity 17 is stronger thn inequlity 2. If x 2n+2 2n+1 we cn find inequlity 2 is stronger thn inequlity 17. The proof of Theorem 3 is completed. 7

8 4 Estimtion Formul of sin x nd cos x THEOREM 4. Let n 1 x π 2 Sn x = sin x x + x3 3! n x 2n 1 2n 1! nd P n x = cos x 1 + x2 2! + + 1n+1 x 2n 2n!. Then x 2n+1 2n + 1! 2n 2n + 1 2n 2n x 2 S n x x2n+1 2n + 1! 2n + 2 2n + 3 2n + 2 2n x n + 2! 2n + 1 2n + 2 2n + 1 2n x 2 P n x x2n+2 2n + 2! 2n + 3 2n + 4 2n + 3 2n x x 2n+2 Proof. Let x π 2 nd f 1 x = sin x + x x3 3! + x4n 3 4n 3! f 2 x = cos x 1 + x2 2! + x4n 2 4n 2! f 3 x = sin x x + x3 3! + x4n 1 4n 1! f 4 x = cos x + 1 x2 2! + + x4n 4n! f 5 x = sin x + x x3 3! + x4n+1 4n + 1! f 6 x = cos x 1 + x2 2! + x4n+2 4n + 2! f 7 x = sin x x + x3 3! + x4n+3 4n + 3! f 8 x = cos x + 1 x2 2! + x4n+4 4n + 4!. We cn find tht sin : x π 2 sin x nd cos : x π 2 cos x re geometriclly concve functions ccording to Lemm 1[5]. By Lemm 2 the following functions ll re geometriclly concve functions on π 2 sin tdt = 1 cos x 1 cos t dt = x sin x t sin t dt = cos x 1 + x2 2. So functions f i i = lso re geometriclly concve functions on π 2. For < < x nd i = Theorem 2 tells us tht Let + hve f i t dt xf i x f i ln x ln. ln xf i x ln f i f i t dt f i+1 x xf i x 1 + lim + xf i x 1 + lim + f i f i f i f i. 23 8

9 With respect to L Hospitl-Rule inequlity 23 becomes So Furthermore f 3 x xf 2 x 4n + 1 f i+1 x f 3 x xf 1 x 4n + 1 f 4 x xf 3 x 4n + 2 xf i x 4n + i f 2 x xf 1 x 4n f 5 x xf 4 x 4n + 3. x2 f 1 x 4n 4n + 1 = x 2 4n 4n + 1 4n 4n + 1 f 3 x x 2 f 3 x + f 3 x xf 3 x 4n + 2 f 4 x 4n + 3 f 5 x = 4n + 3 x x f 3 x + x4n 1 4n 1! x4n+1 4n 1! x4n+1 4n + 1! 4n 4n + 1 4n 4n x f 3 x + x4n+1 4n + 1! x 2 f 3 x 4n + 2 4n + 3 f 3 x + 4n + 2 4n + 3 f 3 x According to 25 nd 26 x 4n+1 4n + 1! 4n 4n + 1 4n 4n x 2 f 3 x Anlogously fter some clculus we cn get x 4n+2 4n + 2! x 4n+3 4n + 3! x 4n+4 4n + 4! x 4n+1 4n + 1! x4n+1 4n + 1! 4n + 2 4n + 3 4n + 2 4n x x4n+1 4n + 1! 4n + 1 4n + 2 4n + 1 4n x 2 f 4 x x4n+2 4n + 2! 4n + 2 4n + 3 4n + 2 4n x 2 f 5 x x4n+3 4n + 3! 4n + 3 4n + 4 4n + 3 4n x 2 f 6 x x4n+4 4n + 4! The proof of Theorem 4 is completed. 4n + 2 4n + 3 4n + 2 4n x n + 3 4n + 4 4n + 3 4n x n + 4 4n + 5 4n + 4 4n x n + 5 4n + 6 4n + 5 4n x 2. 3 REMARK 1. We shll present other pplictions of Theorem 1 nd Theorem 2 in nother pper lso includes estimtion formul of reminder terms in Tylor series expnsion of e x. 9

10 References [1] Xioming Zhng Geometriclly convex function Hefei: An hui University Press 24. Chinese [2] J.Mtkowski L p -like prnorms Selected Topics in Functionl Equtions nd Itertion Theory Proceedings of the Austrin-Polish seminr Grz Mth.Ber. Vol [3] Shijie Li Further results for Jensen s inequlity of convex functions nd pplictions Journl of Fuzhou Techers College Chinese [4] Dinghu Yng About inequlity of geometriclly convex function Hebei University Lerned Journl Nturl Science Edition Chinese [5] Constntin P. Niculescu Convexity ccording tothegeometricmen Mthemticl Inequlities & Applictions [6] Xioming Zhng Some theorem on geometric convex function nd its pplictions Journl of Cpitl Norml University Chinese [7] Xioming Zhng nd Yudong Wu Geometriclly convex functions nd solution of question RGMIA [ONLINE] Avilble online t [8] Ningguo Zheng nd Xioming Zhng An importnt property nd ppliction of geometricl concve functions Mthemtics in Prctice nd Theory Chinese 1