Abstract
Optimal bounds for the weighted geometric mean of the first Seiffert and logarithmic means by weighted generalized Heronian mean are proved. We answer the question: for what the greatest value and the least value such that the double inequality, , holds for all with are. Here, and denote the first Seiffert, logarithmic, and weighted generalized Heronian means of two positive numbers and respectively.
Highlights
Means has been the subject of intensive research
Many remarkable inequalities for the Seiffert, logarithmic, and Heronian mean can be found in the literature [1,2,3,4,5,6,7,8,9,10,11]
In [9], Seiffert proved that L(a, b) < P(a, b) < I(a, b), where I(a, b) is the identric mean
Summary
Optimal bounds for the weighted geometric mean of the first Seiffert and logarithmic means by weighted generalized Heronian mean are proved. P(a, b), L(a, b), and Hω(a, b) denote the first Seiffert, logarithmic, and weighted generalized Heronian means of two positive numbers a and b, respectively
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