裘松良-高校数学教学应充分体现数学发展规律

p A Ny u 5 3 2019 p U M 0 w t n 2019 11 30 t n p A Ny u 5 2019 11 301 21 Jj 1yG 2U M 3 U Y t n p A Ny u 5 2019 11 302 21 yG i 5 5 2015 1 27I 5 w6 o ron f o X t M 7L vkS Uk M IS p I 8vk y3IT u o E u I o 1 L lzc OX k 1 4e 0 2016 4 15 o ron3 AOrN t n p A Ny u 5 2019 11 303 21 yG rn 3 g 3 I f N I 1 e 5r e 0 9 2018 1 3I o ron rNn J M U AO n I N uy Ude 5 0o r U f u J 5 Lu 5K 6 Lu z 7 A o Y d Ny g u 5 K Jp 0 f2 f1 t n p A Ny u 5 2019 11 308 21 U Y eg 2 Let f3 x sinx x for x 0 2 f 0 3 x x cosx sinx x2 f4 x x2 f 0 4 x x sinx 0 f 0 5 x 1 x2 x 1 x log 1 x f6 x x2 f 0 6 x x 1 x 2 0 where h C 0 b diff erentiable on 0 b and on 0 b f 0 7 x 1 x2 xh x Z x 0 h t dt f8 x x2 f 0 8 x xh 0 x 0 f8 f7 resp The computation is sometimes verbose and complicated Common features of the functions in egs 1 4 Recall indetermined forms and l H opital s Rule LR 2 Question 1 Can we fi nd a rule which is similar to LR and makes this kind of study easier t n p A Ny u 5 2019 11 3010 21 U Y ii Analyzing the general case 1 Expected Goal A criterion for the mon properties which is similar to LR 2 In detailed mathematical language For a b 0 resp f x f a g x g a if g x g a g0 x 0 resp f0 y g0 y g x g a g 0 x 0 5 x y a b with y x so that F namely 1 is valid iii An answer to Question 1 Theorem 1 MLR For a b let f g a b R be two continuous functions which are diff erentiable on a b with g0 x 6 0 for x a b If f 0 g0 is resp on a b then so are F x f x f a g x g a and G x f x f b g x g b Moreover mon properties of f 0 g0 is strict so are those of F and G G D Anderson M K Vamanamurthy and M Vuorinen Conformal Invariants Inequalities and Quasiconformal Maps J Wiley Sons 1997 t n p A Ny u 5 2019 11 3013 21 U Y Proof W L G we may assume that f 0 g0 is and g0 x 0 for x a b By Cauchy s Mean Value Thm x a b a x s t F x f x f a g x g a f 0 g0 f 0 x g0 x 6 Clearly g x 0 for all x a b Multiplying 6 by g x g a g0 x we obtain g x g a f 0 x f x f a g0 x 0 F0 x d dx f x f a g x g a 0 x a b 7 yielding the mon property of F The pf of the mon of G is similar Finally if f 0 g0 is strict monotone on a b then both ineqs in 6 and 7 are strict and hence F and G are strictly monotone t n p A Ny u 5 2019 11 3014 21 U Y iv Remark The condition in Thm 1 the mon of f 0 g0 is suffi cient but not necessary v Egs 1 Egs 1 4 aboved mentioned 2 eg 5 f9 x 1 x tanx for x 0 2 f10 x ex 1 x for x 0 3 Appls to the study of convexity concavity properties In eg 2 f3 x 1 x sinx for x 0 2 f 0 3 x x cosx sinx x2 f4 x x2 f 0 4 x x2 0 sinx 2x 1 2f3 x f 0 3 f3is convex t n p A Ny u 5 2019 11 3015 21 U Y In eg 3 f5 x 1 x log 1 x for x 0 f 0 5 x 1 x2 x 1 x log 1 x f6 x x2 f 0 6 x x2 0 1 2 1 x 2 f 0 5 f5is convex vi Open problem Can we establish a similar criterion for the concavity convexity properties of functions of the form f g II Another Criterion for the Mon Properties of Functions Sometimes we need to show the mon properties of functions of the form x P 0 anxn P 0 bnxn for x I an interval Similar to part I Analysis of examples Question Results t n p A Ny u 5 2019 11 3016 21 U Y Theorem 2 S Ponnusamy and M Vuorinen Mathematika 44 1997 Let A x P n 0anx n and B x P n 0bnx n with bn 0 for all n N0 N 0 be two real power series converging on r r If the non constant sequence an bn is in n N0 then the function x A x B x is strictly resp on 0 r Remark A general verersion of Theorem 2 was proved by S Ponnusamy and M Vuorinen in their paper Asymptotic expansions and inequalities for hypergeometric functions Mathematika 1997 44 278 301 t n p A Ny u 5 2019 11 3017 21 U Y Theorem 3 Songliang Qiu Xiaoyan Ma Yuming Chu J Math Anal Appl 474 2019 1306 1337 Suppose that the real power series A x P n 0anx n and B x P n 0bnx n with bn 0 are of a common radius r 0 of convergence and an bn is a non constant sequence Let x A x B x 1 If there is an n0 N such that the sequence an bn is for 0 n n0 and for n n0 then is on 0 r iff 0 r 0 0 r 0 resp 2 If there is an n0 N such that the sequence an bn is for 0 n n0 and for n n0 and if 0 r 0 then there exists a number x0 0 r such that is strictly on 0 x0 and resp on x0 r t n p A Ny u 5 2019 11 3018 21 U Y III A Newest Criterion for the Mon Properties of Functions i Problem Sometimes we need to study the monotonicity properties of the functions of the forms F1 x Z 0 e xtf t dt Z 0 e xtg t dt 1 8 or even more general forms G1 x Z d c f x t dt Z d c g x t dt 1 9 for x a b Question 2 Can we determine the monotonicity properties of the functions in 8 by those of f g in t t n p A Ny u 5 2019 11 3019 21 U Y ii An answer to the Question 2 Theorem 4 QIU 2018 Let f and g be real functions on 0 which have Laplace transformations Z 0 e xtf t dt and Z 0 e xtg t dt and let F1 x R 0 e xtf t dt R 0 e xtg t dt Suppose that g t 6 0