Seiffert Mean Inequality

This paper consists of eight chapters.In the first chapter, we briefly introduce the present situation of the problems which we still study at home and abroad and practical application; also we state the main results obtained in this paper.In the second chapter, putting into use the thinking method of analysis and some tips, we disscuss optimal combination order relation among classical mean arithmetic mean and Heronial dual mean walking with Seiffert mean, and optimal convex and geometric combination boundary among classical mean arithmetic mean and Heronial dual mean walking with Seiffert mean are given.In the third chapter, based on monotonicity of functions and concave convex characteristics analysis and geometric properties, we inquire into optimal combination order relation concerned with Heron mean and Heronial dual mean walking with Seiffert mean, and we get the result that exact convex and geometric combination boundary among Heron mean and Heronial dual mean walking with Seiffert mean.In the fouth chapter, talking over optimal combination order relation among arithmetic mean,geometric mean and the second type Seiffert mean and making use of basic properties of functions and tip of analysis, we draw a conclusion that optimal convex and geometric combination boundary among arithmetic mean and geometric mean walking with Seiffert mean.In the fifth chapter, making researches into optimal combination order relation among arithmetic mean,harmonic mean and the second type Seiffert mean, we will proceed to the conclusion that exact convex and geometric combination boundary among arithmetic mean and harmonic mean walking with the second type Seiffert mean.In the sixth chapter, working at the research of the second type Seiffert mean, deeply discussing optimal combination order relation among arithmetic mean,logarithm mean and the second type Seiffert mean, we arrive at a conclusion that optimal convex and geometric combination bounds of logarithm mean walking with the second type Seiffert mean for arithmetic mean.In the seventh chapter, we proceed to the conclusion that exact convex and geometric combination bounds of Heron mean walking with the second type Seiffert mean for arithmetic mean.In the eighth chapter, we give the conclusion that optimal convex and geometric combination bounds of Heronial dual mean walking with the second type Seiffert mean for the arithmetic mean.