Some More Inequalities for Arithmetic Mean, Harmonic Mean and Variance
نویسنده
چکیده
We derive bounds on the variance of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered, and an operator version is obtained. Some refinements of the Kantorovich inequality are obtained. Bounds for the largest and smallest eigenvalues of a positive definite matrix are also obtained.
منابع مشابه
Two Sharp Inequalities for Bounding the Seiffert Mean by the Arithmetic, Centroidal, and Contra-harmonic Means
In the paper, the authors find the best possible constants appeared in two inequalities for bounding the Seiffert mean by the linear combinations of the arithmetic, centroidal, and contra-harmonic means.
متن کاملOptimal inequalities for the power, harmonic and logarithmic means
For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...
متن کاملInequalities Among Logarithmic-Mean Measures
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented.
متن کاملUpper Bounds on the Probability of Error in terms of Mean Divergence Measures
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, root square mean, etc. Considering the difference of these means, we can establish [5, 6]. some inequalities among them. Interestingly, the difference of mean considered is convex functions. Applying some properties, upper bounds on the probability of error are established in this paper. It is also shown ...
متن کاملPower harmonic aggregation operator with trapezoidal intuitionistic fuzzy numbers for solving MAGDM problems
Trapezoidal intuitionistic fuzzy numbers (TrIFNs) express abundant and flexible information in a suitable manner and are very useful to depict the decision information in the procedure of decision making. In this paper, some new aggregation operators, such as, trapezoidal intuitionistic fuzzy weighted power harmonic mean (TrIFWPHM) operator, trapezoidal intuitionistic fuzzy ordered weighted po...
متن کامل