Some Refinements and Generalizations of I. Schur Type Inequalities

Table 1

Using 𝐻₁(𝑁,𝜆) to approximate the 𝑒^𝜆, when 𝜆=1,3/2,2,3,4 with different 𝑛.

𝑛	𝐻1(𝑛,1)	𝐻1(𝑛,2)	𝐻1(𝑛,3)	𝐻1(𝑛,4)	𝐻1(𝑛,3/2)
	𝑒≈2.71828	𝑒^2≈7.3891	𝑒^3≈20.0855	𝑒^4≈54.5982	𝑒^3/2≈4.481689
1000	2.71828	7.3817	20.0255	54.2724	4.480010
3000	2.71828	7.3866	20.0655	54.4892	4.481129
5000	2.71828	7.3876	20.0735	54.5327	4.481353
7000	2.71828	7.3880	20.0769	54.5514	4.481449
9000	2.71828	7.3882	20.0788	54.5618	4.481502
11000	2.71828	7.3884	20.0801	54.5684	4.481536
13000	2.71828	7.3885	20.0809	54.5730	4.481560
15000	2.71828	7.3886	20.0815	54.5763	4.481577