The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation:

For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

Generalized polynomials are defined as products of polynomials raised to positive real powers. The generalized degree can be introduced in a natural way. Several inequalities holding for ordinary polynomials are expected to be true for generalized polynomials, by utilizing the generalized degree in place of the ordinary one. Based on Remez-type inequalities on the size of generalized polynomials, we establish Bernstein and Markov type inequalities for generalized non-negative polynomials, obtaining the best possible result up to a multiplicative absolute constant.