In the present paper, we answer the question: for 0a what are the greatest value p(a) and the least value q(a) such that the double inequality Jp(a,b)aA(a,b)+ (1-a)G(a,b)Jq(a,b) holds for all a,b>0 with a is not eq...In the present paper, we answer the question: for 0a what are the greatest value p(a) and the least value q(a) such that the double inequality Jp(a,b)aA(a,b)+ (1-a)G(a,b)Jq(a,b) holds for all a,b>0 with a is not equal to?b ?展开更多
This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,&l...This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) and <em>C</em>[<i>λ</i><em>a</em> + (1-<i>λ</i>)<em>b</em>, <i>λ</i><em>b</em> + (1-<i>λ</i>)<em>a</em>] ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <em>C</em>[<i>μ</i><em>a</em> + (1-<i>μ</i>)<em>b</em>, <i>μ</i><em>b</em> + (1-<i>μ</i>)<em>a</em>] with <img src="Edit_ce892b1d-c056-44ea-a929-31dbcd1b0e91.bmp" alt="" /> . The researchers attempt to find an answer as to what are the best possible parameters <i>λ</i>, <i>μ</i> that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert <em>f</em>(<i>t</i>) = <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_efa43881-9a60-44f8-a86f-d4a1057f4378.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become f (<i>t</i>) ≤ 0. Secondly, we insert g(<i>t</i>) = <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_750dddbb-1d71-45d3-be29-6da5c88ba85d.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become <em>g</em>(<i>t</i>) ≥ 0.展开更多
In this paper, we considered the equality problem of weighted Bajraktarević means with weighted quasi-arithmetic means. Using the method of substituting for functions, we first transform the equality problem into solv...In this paper, we considered the equality problem of weighted Bajraktarević means with weighted quasi-arithmetic means. Using the method of substituting for functions, we first transform the equality problem into solving an equivalent functional equation. We obtain the necessary and sufficient conditions for the equality equation.展开更多
In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2)a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),...In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2)a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2)sin^(2))1/2 tdt.展开更多
In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logari...In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.展开更多
In the present paper, we answer the question: for 0 what are the greatest value p(a) and the least value q(a) such that the inequality. For more information about abstract,please download the PDF file.
The main purpose of this paper is using the analytic method to study the mean value properties of the arithmetical functions δk（（m, n））, δk（[m,n]/m）,and give several interesting asymptotic formulae for them.
This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtaine...This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.展开更多
文摘In the present paper, we answer the question: for 0a what are the greatest value p(a) and the least value q(a) such that the double inequality Jp(a,b)aA(a,b)+ (1-a)G(a,b)Jq(a,b) holds for all a,b>0 with a is not equal to?b ?
文摘This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) and <em>C</em>[<i>λ</i><em>a</em> + (1-<i>λ</i>)<em>b</em>, <i>λ</i><em>b</em> + (1-<i>λ</i>)<em>a</em>] ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <em>C</em>[<i>μ</i><em>a</em> + (1-<i>μ</i>)<em>b</em>, <i>μ</i><em>b</em> + (1-<i>μ</i>)<em>a</em>] with <img src="Edit_ce892b1d-c056-44ea-a929-31dbcd1b0e91.bmp" alt="" /> . The researchers attempt to find an answer as to what are the best possible parameters <i>λ</i>, <i>μ</i> that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert <em>f</em>(<i>t</i>) = <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_efa43881-9a60-44f8-a86f-d4a1057f4378.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become f (<i>t</i>) ≤ 0. Secondly, we insert g(<i>t</i>) = <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_750dddbb-1d71-45d3-be29-6da5c88ba85d.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become <em>g</em>(<i>t</i>) ≥ 0.
文摘In this paper, we considered the equality problem of weighted Bajraktarević means with weighted quasi-arithmetic means. Using the method of substituting for functions, we first transform the equality problem into solving an equivalent functional equation. We obtain the necessary and sufficient conditions for the equality equation.
基金supported by the Natural Science Foundation of China(61673169,11301127,11701176,11626101,11601485)。
文摘In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2)a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2)sin^(2))1/2 tdt.
基金国家自然科学基金(61640020)江苏省农业自主创新项目(CX(13)3054+5 种基金CX(16)1006)江苏省重点研发计划(BE2016368-1)江苏省科技重点及面上项目(SBE2018310371)弹总装线***技术研究(JCKY2017***)Postgraduate Research&Practice Innovation Program of Jiangsu Province(SJCX17_0107)北京市教育委员会科技计划面上项目(KM201510028019)
文摘In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.
文摘In the present paper, we answer the question: for 0 what are the greatest value p(a) and the least value q(a) such that the inequality. For more information about abstract,please download the PDF file.
基金Supported by NSF of China(10671155)Supported by SF of Education Department of Shannxi Province(08JK291)
文摘The main purpose of this paper is using the analytic method to study the mean value properties of the arithmetical functions δk（（m, n））, δk（[m,n]/m）,and give several interesting asymptotic formulae for them.
文摘This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.
