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Analysis 22, 277 - 283 (2002) Analysis © Oldenbourg Verlag München 2002 THE ARITHMETIC MEAN - GEOMETRIC MEAN INEQUALITY FOR COMPLEX NUMBERS Horst Alzer and Stephan Ruscheweyh' Received: February 5 , 2002 Abstract. We prove: let (p € (0,7r/2) and W^ = {z e C : \aigz\ < (f>}. Then we have for all Zi,...,Zn e W^: (Ol) n k , | ' / " < a ( i [ ^ ] , c o s ( 2 0 ) ) i f ^ z , where - 1) + - ß) + a ^ l - 2(1^)" = 7 ^ (2(1 - m) + - 1) + a ^ / M l - fi) + a ^ l - 2 f iy j The given constant factor is best possible for every n, <j). Our result extends a

Abstract

In this paper we obtain some trace inequalities for positive operators via recent refinements and reverses of Young’s inequality due to Kittaneh-Manasrah, Liao-Wu-Zhao, Zuo-Shi-Fujii, Tominaga and Furuichi.

Abstract

In this paper we establish some inequalities of Hermite-Hadamard type for operator convex functions and positive maps. Applications for power function and logarithm are also provided.

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Analysis Volume 22 • Issue 3 • 2002 M. Obradovic, S. Ponnusamy, V. Singh and P. Vasundhra Univalency, starlikeness and convexity applied to certain classes of rational functions 225 B. Fritzsche, B. Kirstein and A. Lasarow On Schwarz-Pick-Potapov block matrices of matrix-valued functions which belong to the extended Potapov class 243 Mursaleen and O. H. H. Edely On some Statistical core theorems 265 H. Alzer and S. Ruscheweyh The arithmetic mean - geometric mean inequality for complex numbers 277 R. GarunkMs, A. Laurindikas, R. Slezeviöiene, J

.— YANG, Z. H.: Mixed mean inequalities for several positive definite matrices, Linear Algebra Appl. 395 (2005), 247–263. http://dx.doi.org/10.1016/j.laa.2004.08.010 [6] KEDLAYA, K.: Proof of a mixed arithmetic-mean, geometric-mean inequality, Amer. Math. Monthly 101 (1994), 355–357. http://dx.doi.org/10.2307/2975630 [7] LENG, G. S.— SI, L.— ZHU, Q. S.: Mixed-mean inequalities for subsets, Proc. Amer. Math. Soc. 132 (2004), 2655–2660. http://dx.doi.org/10.1090/S0002-9939-04-07384-8 [8] MATSUDA, T.: An inductive proof of a mixed arithmetic-geometric mean inequality, Amer

Author Index of Volume 22 (2002) Alzer, H., Ruscheweyh, S., The arithmetic mean - geometric mean inequality for complex numbers 278 Anbudurai, M., Parvatham, R., Ponnusamy, S., Singh, V., Duality for Hadamard products applied to functions satisfying certain differential inequalities 381 Begehr, Η., Qin, D., On the Riemann-Hilbert-Poincare problem for analytic functions 183 Rusev,P.,. see Boychev, G. 67 Bradley, D. M., Some remarks on sine integrals and their connection with combinatorics, geo- metry and probability 219 Calderon-Moreno, M. C., Universal

boundary 201 D. M. Bradley Some remarks on sine integrals and their connection with combinatorics, geometry and probability 219 Number 3 M. Obradovic, S. Ponnusamy, V. Singh and P. Vasundhra Univalency, starlikeness and convexity applied to certain classes of rational functions 225 B. Fritzsche, B. Kirstein and A. Lasarow On Schwarz-Pick-Potapov block matrices of matrix-valued functions which belong to the extended Potapov class 244 Mursaleen and Ο. Η. H. Edely On some statistical core theorems 266 H. Alzer and S. Ruscheweyh The arithmetic mean - geometric

, s ≤ t {s\leq t} , we obtain (2.3) x ⁢ ( τ i ⁢ ( s ) ) = x ⁢ ( t ) ⁢ exp ⁡ ( ∫ τ i ⁢ ( s ) t ∑ ℓ = 1 m p ℓ ⁢ ( u ) ⁢ x ⁢ ( τ ℓ ⁢ ( u ) ) x ⁢ ( u ) ⁢ d ⁢ u ) , i = 1 , 2 , … , m . x(\tau_{i}(s))=x(t)\exp\Biggl{(}\int_{\tau_{i}(s)}^{t}\sum_{\ell=1}^{m}p_{\ell% }(u)\frac{x(\tau_{\ell}(u))}{x(u)}\mathop{}\!du\Biggr{)},\quad i=1,2,\dots,m. Using the fact that x ⁢ ( τ ℓ ⁢ ( u ) ) x ⁢ ( u ) ≥ 1 {\frac{x(\tau_{\ell}(u))}{x(u)}\geq 1} and the arithmetic mean-geometric mean inequality, ( 2.3 ) leads to (2.4) x ⁢ ( τ i ⁢ ( s ) ) ≥ x ⁢ ( t ) ⁢ exp ⁡ ( ∫ τ i ⁢ ( s ) t m

[ln (G(f(x), f(a+b- *)))] on [a, b] and using the well-known Jensen's integral inequality for the convex mapping exp(-), we have Inequalities of Hadamard type 359 i * -¡—\G(f(x),f(a + b-x))dx a i * = S exp[ln(G(/(x), f(a + b - *)))] dx b — a J > exp = exp = exp 1 6 — \]n[G(f(z),f(a+b-x))]dx a " 1 t /an/ (x ) + ln(/(a + 6 - x ) ) \ 1 b-a H 2 L n N 0 —— I In / (x ) dx — a J since obviously J In / (x ) dx = J In / (a + b — x) dx. So the second inequality in (2.8) is proved. By the arithmetic mean-geometric mean inequality, we have that

INDEX Activation/inhibition: equations for, 318; mechanisms of, 318 Aeolian tones, produced by wires and trees, 286 Aerosols, 78; height of layer, 72 Air mass (AM), 66 Airy differential equation, 279; functions, 279; integral, 92; theory, 9 Allometric growth, 48 Ants, giant and normal size, 43 Archimedes, bounds on π , 235 Archimedes’ principle, 34 Arithmetic mean/geometric mean inequality, 24 Armstrong, W., 130 Asteroid, 21 Atmosphere: thickness of, 66, 73; weight of, 23 Atomic explosion, shock front produced by, 53 Ball, P., 190, 320 “Band” tectonics, 25 Bay of