Sharp Inequalities Involving Neuman Means of the Second Kind with Applications

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1 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly Sharp Ineqalities Involving Neman Means of the Second Kind with Applications Lin-Chang Shen, Ye-Ying Yang * Wei-Mao Qian 3 Hzho Shanlian Adlt School,Hzho, Zhejiang,China Mechanic Electronic Atomobile Egineering College, Hzho Vocational & Technical College, Hzho, Zhejiang,China 3 School of Distance Edcation, Hzho Broadcast TV University, Hzho, Zhejiang,China yyy008hz@63.com. Abstract In this paper, we give the eplicit formlas for Neman means of the second kind N GQ(a, b) N QG(a, b), find the best possible parameters α i, β i (0, )(i =,, 3,, 6) sch that the doble ineqalities α Q(a, b) + ( α )G(a, b) < N QG(a, b) < β Q(a, b) + ( β )G(a, b), α G(a, b) + α Q(a, b) < N QG(a, b) < β G(a, b) + β Q(a, b), α 3Q(a, b) + ( α 3)G(a, b) < N GQ(a, b) < β 3Q(a, b) + ( β 3)G(a, b), α 4 G(a, b) + α4 Q(a, b) < N GQ(a, b) < β4 G(a, b) + β4 Q(a, b), α 5Q(a, b) + ( α 5)V (a, b) < N QG(a, b) < β 5Q(a, b) + ( β 5)V (a, b), α 6Q(a, b) + ( α 6)U(a, b) < N GQ(a, b) < β 6Q(a, b) + ( β 6)U(a, b), holds for all a, b > 0 with a b, where G(a, b) Q(a, b) are the classical geometric qadratic means, V (a, b), U(a, b), N QG(a, b) N GQ(a, b) are Yang Neman mean of the second kind. Keywords: geometric mean, qadratic mean, Neman means of the second kind, Yang means, ineqalities. Introdction For a, b > 0 with a b, the Schwab-Borchardt mean SB(a, b), is defined by b a cos (a/b), if a < b, SB(a, b) = cosh (a/b), if a > b. where cos () cosh () = log( + ) are the inverse cosine inverse hyperbolic cosine fnctions, respectively. It is well-known that SB(a, b) is strictly increasing in both a b, nonsymmetric homogeneos of degree with respect to a b. Many symmetric bivariate means are special cases of the Schwab- Borchardt mean, for eample, the first second Seiffert means, Neman-Sándor mean, logarithmic mean two Yang means 3 are respectively defined by P = P (a, b) = sin = SB(G, A), ()/(a + b) T = T (a, b) = tan = SB(A, Q), ()/(a + b) Copyright 06 Isaac Scientific Pblishing

2 40 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly 06 M = M(a, b) = sinh = SB(Q, A), ()/(a + b) L = L(a, b) = tanh = SB(A, G), ()/(a + b) U = U(a, b) = tan ()/ = SB(G, Q), () ab V = V (a, b) = sinh ()/ = SB(Q, G). () ab where G = G(a, b) = ab, A = A(a, b) = (a + b)/ Q = Q(a, b) = (a + b )/ are the classical geometric, arithmetic qadratic means of a b. Let X = X(a, b) Y = Y (a, b) be the symmetric bivariate means of a b. Then Neman mean of the second kind N XY (a, b)4 is defined by N XY (a, b) = Y X +. (3) SB(X, Y ) Moreover, withot loss of generality, let a > b, v = ()/(a + b) (0, ), then Neman 4 gave eplicit formlas N AG (a, b) = A + ( v ) tanh (v) v N AQ (a, b) = A + ( + v ) tan (v) v ineqalities, N GA (a, b) = A v + sin (v) v, N QA (a, b) = A + v + sinh (v) v G(a, b) < L(a, b) < N AG (a, b) < P (a, b) < N GA (a, b) < A(a, b) < M(a, b) < N QA (a, b) < T (a, b) < N AQ (a, b) < Q(a, b). for all a, b > 0 with a b. In the recent past, the Schwab-Borchardt mean has been the sbject of intensive research. In particlar, many remarkable ineqalities for Schwab-Borchardt mean its generated means can be fond in the literatre 4-4. In 4, Neman fond the best possible constants α, α, α 3, α 4 β, β, β 3, β 4 sch that the doble ineqalities α A(a, b) + ( α )G(a, b) < N GA (a, b) < β A(a, b) + ( β )G(a, b) α Q(a, b) + ( α )A(a, b) < N AQ (a, b) < β Q(a, b) + ( β )A(a, b) α 3 A(a, b) + ( α 3 )G(a, b) < N AG (a, b) < β 3 A(a, b) + ( β 3 )G(a, b) α 4 Q(a, b) + ( α 4 )A(a, b) < N QA (a, b) < β 4 Q(a, b) + ( β 4 )A(a, b) hold for a, b > 0 with a b if only if α /3, β π/4, α /3, β (π )/ 4( ) = 0.689, α 3 /3, β 3 / α 4 /3, β 4 log( + ) + / ( ) = Zhang et al. presented the best possible parameters α, α, β, β 0, / α 3, α 4, β 3, β 4 /, sch that the doble ineqalities G ( α a + ( α )b, α b + ( α )a ) < N AG (a, b) < G ( β a + ( β )b, β b + ( β )a ) G ( α a + ( α )b, α b + ( α )a ) < N GA (a, b) < G ( β a + ( β )b, β b + ( β )a ) Q ( α 3 a + ( α 3 )b, α 3 b + ( α 3 )a ) < N QA (a, b) < Q ( β 3 a + ( β 3 )b, β 3 b + ( β 3 )a ) Q ( α 4 a + ( α 4 )b, α 4 b + ( α 4 )a ) < N AQ (a, b) < Q ( β 4 a + ( β 4 )b, β 4 b + ( β 4 )a ). Copyright 06 Isaac Scientific Pblishing

3 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly 06 4 hold for all a, b > 0 with a b. Go et.al. proved that the doble ineqalities A p (a, b)g p (a, b) < N GA (a, b) < A q (a, b)g q (a, b), p G(a, b) + p A(a, b) < N GA(a, b) < q G(a, b) + q A(a, b), A p3 (a, b)g p3 (a, b) < N AG (a, b) < A q3 (a, b)g q3 (a, b), p 4 G(a, b) + p 4 A(a, b) < N AG(a, b) < q 4 G(a, b) + q 4 A(a, b), Q p5 (a, b)a p5 (a, b) < N AQ (a, b) < Q q5 (a, b)a q5 (a, b), p 6 A(a, b) + p 6 Q(a, b) < N AQ(a, b) < q 6 A(a, b) + q 6 Q(a, b), Q p7 (a, b)a p7 (a, b) < N QA (a, b) < Q q7 (a, b)a q7 (a, b), p 8 A(a, b) + p 8 Q(a, b) < N QA(a, b) < q 8 A(a, b) + q 8 Q(a, b). hold for all a, b > 0 with a b if only if p /3, q, p 0, q /3, p 3 /3, q 3, p 4 0, q 4 /3, p 5 /3, q 5 log(π+)/ log 4 = 0.744, p 6 6+ (+ )π /(π+) = 0.49, q 6 /3, p 7 /3, q 7 log +log(+ ) / log = p 8 + (+ ) log(+ ) / + log( + ) = , q8 /3. Let a > b > 0, = ()/ ab (0, + ). Then from ()-(3) we gave the eplicit formlas N QG (a, b) = G(a, b) + + sinh (). (4) N GQ (a, b) = G(a, b) + ( + ) tan (). (5) The main prpose of this paper is to find the best possible parameters α i, β i (0, )(i =,, 3,, 6) sch that the doble ineqalities hold for all a, b > 0 with a b. α Q(a, b) + ( α )G(a, b) < N QG (a, b) < β Q(a, b) + ( β )G(a, b), α G(a, b) + α Q(a, b) < N QG (a, b) < β G(a, b) + β Q(a, b), α 3 Q(a, b) + ( α 3 )G(a, b) < N GQ (a, b) < β 3 Q(a, b) + ( β 3 )G(a, b), α 4 G(a, b) + α 4 Q(a, b) < N GQ (a, b) < β 4 G(a, b) + β 4 Q(a, b), α 5 Q(a, b) + ( α 5 )V (a, b) < N QG (a, b) < β 5 Q(a, b) + ( β 5 )V (a, b), α 6 Q(a, b) + ( α 6 )U(a, b) < N GQ (a, b) < β 6 Q(a, b) + ( β 6 )U(a, b). Lemma In order to prove or main reslts we need several lemmas, which we present in this section. Lemma. (see5) For < a < b < +, let f, g : a, b R be continos on a, b, be differentiable on (a, b), let g () 0 on (a, b). If f ()/g () is increasing (decreasing) on (a, b), then so are f() f(a) g() g(a), f() f(b) g() g(b) If f ()/g () is strictly monotone, then the monotonicity in the conclsion is also strict. Copyright 06 Isaac Scientific Pblishing

4 4 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly 06 Lemma. (see 6). Sppose that the power series f() = a n n g() = b n n have the radis of convergence r > 0 with a n, b n > 0 for all n = 0,,,. Let h() = f()/g(), if the seqence series {a n /b n } is (strictly) increasing (decreasing), then h() is also (strictly) increasing (decreasing) on (0, r). Lemma.3 () (See 7, Lemma.4) The fnction ϕ () = is strictly increasing from (0, + ) onto(/3, ). ()(See 7, Lemma.6) The fnction ϕ () = + sinh() 4 sinh() sinh() sinh() sinh() cosh() cosh() + sinh() cosh() is strictly decreasing from (0, + ) onto (0, /3). (3)(See 7, Lemma.5) The fnction ϕ 3 () = is strictly increasing from (0, π/) onto(8/3, π). (4)(See 7, Lemma.8) The fnction ϕ 4 () = sin() sin() cos() sin() cos() cos() + sin() cos() is strictly decreasing from (0, π/) onto(, /3). Lemma.4 The fnction ϕ 5 () = sinh() sinh() cosh() + is strictly decreasing from (0, + ) onto(, ). Proof. Let f () = sinh(), g () = sinh() cosh() +. Then simple comptations lead to ϕ 5 () = f () g () = f () f (0 + ) g () g (0 + ). (6) Let f () sinh() + cosh() 4 g = () cosh() sinh() n (n)! n + = n (n)! n = a n = n= n= (n+) n+ (n+)! n+ n+ (n+)! n+ 4 n n+ (n+)! n+ = (n + ) n+4 (n + 3)! > 0, b n = n+ (n+)! n+ (n+) n+4 (n+3)! n (n+) n+4 (n+3)! n. (n + ) n+4 (n + 3)! (7) > 0. (8) a n+ a n = b n+ b n (n + )(n + ) < 0. (9) Copyright 06 Isaac Scientific Pblishing

5 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly for all n 0. It follows from Lemma. (7)-(9) that f ()/g () is strictly decreasing on (0, + ). Note that ϕ 5 (0 + ) = a 0 b 0 =, ϕ 5 (+ ) =. (0) Therefore, Lemma.4 follows easily from Lemma. (6), (0) together with the monotonicity of f ()/g (). Lemma.5 The fnction ϕ 6 () = + sin() cos() sin () sin() sin() is strictly increasing from (0, π/) onto (0, (π 8)/(π )). Proof.The fnction ϕ 6 () can be rewritten as ϕ 6 () = + cos() sin() + = ϕ 7 () + ϕ 8 (). () sin() sin() where ϕ 7 () = / sin() ϕ 8 () = + cos() sin()/ sin(). Let f () = + cos() sin(), g () = sin(), f 3 () = cos() sin() g 3 () = cos(). Then simple comptations lead to ϕ 8 () = f () g () = f () f (0 + ) g () g (0 + ). () f () g () = f 3() g 3 () = f 3() f 3 (0 + ) g 3 () g 3 (0 + ). (3) f 3() g 3 () = tan(). (4) Since the fnction / tan() is strictly decreasing on (0, π/), hence Lemma. ()-(4) lead to that ϕ 8 () is strictly increasing on (0, π/). From () the fact that the fnction ϕ 7 () = / sin() is strictly increasing on (0, π/) together with the monotonicity of ϕ 8 () we can reach the conclsion that ϕ 6 () is strictly increasing on (0, π/). Note that ϕ 6 (0 + ) = 0, ϕ 6 ( π ) = π 8 (π ). (5) Therefore, Lemma.5 follows easily from (5) the monotonicity of ϕ 6 (). 3 Main Reslts Theorem 3. The doble ineqalities α Q(a, b) + ( α )G(a, b) < N QG (a, b) < β Q(a, b) + ( β )G(a, b). (6) α G(a, b) + α Q(a, b) < N QG (a, b) < β G(a, b) + β Q(a, b). (7) hold for all a, b > 0 with a b if only if α /3, β /, α 0 β /3. Proof.We clearly see that ineqalities (6) (7) can be rewritten as α < N QG(a, b) G(a, b) Q(a, b) G(a, b) < β. (8) α < /N QG(a, b) /Q(a, b) /G(a, b) /Q(a, b) < β. (9) Copyright 06 Isaac Scientific Pblishing

6 44 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly 06 respectively. Since both the geometric mean G(a, b) qadratic mean Q(a, b) are symmetric homogeneos of degree, withot loss of generality, we assme that a > b > 0. Let = ()/ ab (0, + ). Then from (4) (8)-(9) together with Q(a, b) = G(a, b) + we have α < + + sinh () + < β. (0) + α < sinh () ( + ) < β. () + + sinh () respectively. Let = sinh (). Then (0, + ), + + sinh () + = + sinh() 4 sinh() = sinh() sinh() ϕ (). () + sinh () ( + ) + + sinh () = sinh() cosh() cosh() + sinh() cosh() = ϕ (). where the fnctions ϕ () ϕ () are defined as in Lemma.3() (). Therefore, ineqality (6) holds for all a, b > 0 with a b if only if α /3 β / follows from (0) () together with Lemma.3(), ineqality (7) holds for all a, b > 0 with a b if only if α 0 β /3 follows from () (3) together with Lemma.3(). Theorem 3. The doble ineqalities α 3 Q(a, b) + ( α 3 )G(a, b) < N GQ (a, b) < β 3 Q(a, b) + ( β 3 )G(a, b). (4) (3) α 4 G(a, b) + α 4 Q(a, b) < N GQ (a, b) < β 4 G(a, b) + β 4 Q(a, b). (5) holds for all a, b > 0 with a b if only if α 3 /3, β 3 π/4, α 4 0 β 4 /3. Proof.We clearly see that ineqalities (4) (5) can be rewritten as α 3 < N GQ(a, b) G(a, b) Q(a, b) G(a, b) α 4 < /N GQ(a, b) /Q(a, b) /G(a, b) /Q(a, b) < β 3. (6) < β 4. (7) respectively. Since both the geometric mean G(a, b) qadratic mean Q(a, b) are symmetric homogeneos of degree, withot loss of generality, we assme that a > b > 0. Let = ()/ ab (0, + ). Then from (5) (6)-(7) together with Q(a, b) = G(a, b) + we have α 3 < + ( + ) tan () + < β 3. (8) Copyright 06 Isaac Scientific Pblishing

7 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly α 4 < + + ( + ) tan () ( < β 4. (9) + ) + ( + ) tan () respectively. Let = tan (). Then (0, π/), + ( + ) tan () + = sin() 4 sin() cos() = 4 ϕ 3(). + + ( + ) tan () ( + ) + ( + ) tan () = + sin() cos() cos() + sin() cos() = + ϕ 4(). (30) (3) where the fnctions ϕ 3 () ϕ 4 () are defined as in Lemma.3(3).3(4). Therefore, ineqality (4) holds for all a, b > 0 with a b if only if α 3 /3 β 3 π/4 follows from (8) (30) together with Lemma.3(3), ineqality (5) holds for all a, b > 0 with a b if only if α 4 0 β 4 /3 follows from (9) (3) together with Lemma.3(4). Theorem 3.3 The doble ineqalities α 5 Q(a, b) + ( α 5 )V (a, b) < N QG (a, b) < β 5 Q(a, b) + ( β 5 )V (a, b). (3) holds for all a, b > 0 with a b if only if α 5 0 β 5 /. Proof.We clearly see that ineqalities (3) can be rewritten as α 5 < N QG(a, b) V (a, b) Q(a, b) V (a, b) < β 5. (33) Since both the geometric mean G(a, b) qadratic mean Q(a, b) are symmetric homogeneos of degree, withot loss of generality, we assme that a > b > 0. Let = ()/ ab (0, + ). Then from (4) (33) together with Q(a, b) = G(a, b) + we have α 5 < + + sinh () + sinh () sinh () < β 5. (34) Let = sinh (). Then (0, + ), + + sinh () + sinh () sinh () = sinh() sinh() cosh() + = ϕ 5(). (35) where the fnctions ϕ 5 () is defined as in Lemma.4. Therefore, ineqality (3) holds for all a, b > 0 with a b if only if α 5 0 β 5 / follows from (34) (35) together with Lemma.4. Copyright 06 Isaac Scientific Pblishing

8 46 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly 06 Theorem 3.4 The doble ineqalities α 6 Q(a, b) + ( α 6 )U(a, b) < N GQ (a, b) < β 6 Q(a, b) + ( β 6 )U(a, b). (36) holds for all a, b > 0 with a b if only if α 6 0, β 6 (π 8)/4(π ) = Proof.We clearly see that ineqalities (36) can be rewritten as α 6 < N GQ(a, b) U(a, b) Q(a, b) U(a, b) < β 6. (37) Since both the geometric mean G(a, b) qadratic mean Q(a, b) are symmetric homogeneos of degree, withot loss of generality, we assme that a > b > 0. Let = ()/ ab (0, + ). Then from (5) (36) together with Q(a, b) = G(a, b) + we have α 6 < Let = tan (). Then (0, π/), + ( + ) tan () + + ( + ) tan () + tan () tan () tan () tan () < β 6. (38). = ϕ 6(), (39) where the fnction ϕ 6 () is defined as in Lemma.5. Therefore, ineqality (36) holds for all a, b > 0 with a b if only if α 6 0 β 6 (π 8)/4(π ) = follows from (37)-(39) together with Lemma.5. 4 Applications In this section, we will establish several sharp ineqalities involving the hyperbolic, inverse hyperbolic, trigonometric inverse trigonometric fnctions by se of Theorems From (3) we clearly see that N QG (a, b) = Q(a, b) + G (a, b) V (a, b), N GQ (a, b) = G(a, b) + Q (a, b) U(a, b) Let a > b = sinh ( a b ab ) (0, ). Then simple comptations lead to Q(a, b) V (a, b) = cosh(), G(a, b) G(a, b) = sinh() Theorems (40)-(4) lead to Theorem 4.. Theorem 4. The doble ineqalities α cosh() + ( α ) < cosh() +,. (40) U(a, b) G(a, b) = sinh() tan. (4) sinh() sinh() < β cosh() + ( β ), α cosh() + ( α ) < sinh() + < β cosh() + ( β ), α 3 cosh() + ( α 3 ) < cosh() coth() tan sinh() < β 3 cosh() + ( β 3 ), α 4 cosh() + ( α 4 ) cosh() < + cosh() coth() tan sinh() < β 4 cosh() + ( β 4 ), cosh() Copyright 06 Isaac Scientific Pblishing

9 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly α 5 cosh() + ( α 5 ) sinh() < cosh() + sinh() < β 5 cosh() + ( β 5 ) sinh(), sinh() α 6 cosh() + ( α 6 ) tan sinh() < + cosh() coth() tan sinh() sinh() < β 6 cosh() + ( β 6 ) tan sinh(). hold for all > 0 if only if α /3, β /, α 0, β /3, α 3 /3, β 3 π/4, α 4 0, β 4 /3, α 5 0, β 5 /, α 6 0 β 6 (π 8)/ 4(π ). Let a > b = tan ( ) a b ab (0, π/). Then it is not difficlt to verify that Q(a, b) V (a, b) = sec(), G(a, b) G(a, b) = From Theorems (40), (4) we get Theorem 4. immediately. Theorem 4. The doble ineqalities α sec() + ( α ) < sec() + sinh tan() tan() sinh U(a, b), tan() G(a, b) = tan(). (4) tan() < β sec() + ( β ), α + ( α ) cos() tan() < sec() tan() + sinh tan() < β + ( β ) cos(), α 3 sec() + ( α 3 ) < + sin() < β 3 sec() + ( β 3 ), α4 + ( α 4 ) cos() < sin() + < β4 + ( β 4 ) cos(), tan() α 5 sec()+( α 5 ) sinh tan() < sec()+ sinh tan() tan() < β 5 sec()+( β 5 ) tan() sinh tan(), α 6 sec() + ( α 6 ) tan() < + sin() < β 6 sec() + ( β 6 ) tan(). hold for all (0, π/) if only if α /3, β /, α 0, β /3, α 3 /3, β 3 π/4, α 4 0, β 4 /3, α 5 0, β 5 /, α 6 0 β 6 (π 8)/ 4(π ). Acknowledgments. This research was spported by the Natral Science Fondation of Zhejiang Broadcast TV University nder Grant XKT-5G7. References. E. Neman, J. Sándor, "On the Schwab-Borchardt mean", Math. Pann., vol.4, no., pp.53-66, E. Neman, J. Sándor, "On the Schwab-Borchardt mean II", Math. Pann., vol.7, no., pp.49-59, Z.-H.Yang, "Three families of two-parameter means constrcted by trigonometric fnctions", J. Ineqal. Appl., 54, 7 pages(03). 4. E. Neman, "On a new bivariate mean", Aeqat. Math., vol.88, no.3, pp.77-89, E. Neman, "On some means derived from the Schwab-Borchardt mean", J. Math. Ineqal., vol.8, pp., pp.7-83, E. Neman, "On some means derived from the Schwab-Borchardt mean II", J. Math. Ineqal.,vol.8, pp., , Y.-M. Ch, W.-M. Qian, L.-M. W, X.-H. Zhang, "Optimal bonds for the first second Seiffert means in terms of geometric, arithmetic contraharmonic means", J. Ineqal. Appl.,44(05). 8. Y.-M. Ch, W.-M. Qian, "Refinements of bonds for Neman means, Abstr". Appl. Anal., Article ID 543, 8 pages (004). Copyright 06 Isaac Scientific Pblishing

10 48 Jornal of Advances in Applied Mathematics, Vol., No. 3, Jly W.-M. Qian, Y.-M. Ch, "Optimal bonds for Neman means in terms of geometric, arithmetic qadratic means", J. Ineqal. Appl., 04:75 (04). 0. W.-M. Qian, Z.-H Shao Y.-M. Ch, "Sharp Ineqalities Involving Neman Means of the Second Kind", J. Math. Ineqal., vol.9, no. pp , 05.. Y. Zhang, Y.-M. Ch Y.-L. Jiang, "Sharp geometric mean bonds for Nemam means", Abstr. Appl. Anal., Article ID94988, 6 pages, (04).. Z.-J.Go, Y. zhang, Y.- M. Ch Y.-Q. Song, "Sharp bonds for Neman means in terms of geometric, arithmetic qadratic means", Available online at ariv. org/abs/ Y.-Y. Yang, W.-M. Qian, "The optimal conve combination bonds of harmonic, arithmetic contraharmonic means for the Neman means", Int. Math. Form, vol.9, no.7, pp , Z.-Y. He, Y.-M. Ch M.-K. Wang, "Optimal bonds for Neman means in terms of harmonic contraharmonic means", J. Appl. Math., Article ID 80763, 4 pages(03). 5. G. D. Anderson, M. K. Vamanamrthy M. K. Vorinen, Conformal Invariants, Ineqalities, Qasiconformal Maps, Canadian Mathematical Society Series of Monographs Advanced Tets, John Wiley & Sons, New York, NY, USA, S. Simić M. Vorinen, "Len ineqalities for zero-balanced hypergeometric fnction", Abstr. Appl. Anal., Article ID 9306, pages(0). 7. S.- B. Chen, Z.-Y. He, Y.-M. Ch, Y.-Q. Song X.-J. Tao, "Note on certain ineqalities for Neman means", J. Ineqal. Appl., 04:370(04). Copyright 06 Isaac Scientific Pblishing