On strengthened weighted Carleman's inequality

In this paper we prove a new refinement of the weighted arithmetic-geometric mean inequality and apply this result in obtaining a sharpened version of the weighted Carleman's inequality.

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On the Means of Jordan’s Totient Function

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-020-00356-y ◽

2020 ◽

Vol 46 (6) ◽

pp. 1753-1765

Author(s):

Mohammadreza Esfandiari

Keyword(s):

Asymptotic Formula ◽

Geometric Mean ◽

Harmonic Mean ◽

Alternating Sums ◽

Important Means ◽

Carleman’S Inequality ◽

Carleman's Inequality

Abstract In this paper, we study some important means of Jordan’s totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan’s totient function and Carleman’s inequality for this function.

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The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽

10.3390/sym10090380 ◽

2018 ◽

Vol 10 (9) ◽

pp. 380 ◽

Cited By ~ 2

Author(s):

Yongtao Li ◽

Xian-Ming Gu ◽

Jianxing Zhao

Keyword(s):

Geometric Mean ◽

Arithmetic Mean ◽

Hölder Inequality ◽

Power Mean ◽

Weighted Arithmetic ◽

Mathematical Equivalence ◽

Power Mean Inequality ◽

Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

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Geospatial Suitability Indices Toolbox (GSI Toolbox)

10.21079/11681/41881 ◽

2021 ◽

Author(s):

Christina Saltus ◽

Todd Swannack ◽

S. McKay

Keyword(s):

Habitat Suitability ◽

Geometric Mean ◽

Arithmetic Mean ◽

Limiting Factor ◽

Ecological Processes ◽

Weighted Arithmetic ◽

Suitability Index ◽

Multiple Parameter ◽

Spatially Distributed

Habitat suitability models are widely adopted in ecosystem management and restoration, where these index models are used to assess environmental impacts and benefits based on the quantity and quality of a given habitat. Many spatially distributed ecological processes require application of suitability models within a geographic information system (GIS). Here, we present a geospatial toolbox for assessing habitat suitability. The Geospatial Suitability Indices (GSI) toolbox was developed in ArcGIS Pro 2.7 using the Python® 3.7 programming language and is available for use on the local desktop in the Windows 10 environment. Two main tools comprise the GSI toolbox. First, the Suitability Index Calculator tool uses thematic or continuous geospatial raster layers to calculate parameter suitability indices based on user-specified habitat relationships. Second, the Overall Suitability Index Calculator combines multiple parameter suitability indices into one overarching index using one or more options, including: arithmetic mean, weighted arithmetic mean, geometric mean, and minimum limiting factor. The resultant output is a raster layer representing habitat suitability values from 0.0 to 1.0, where zero is unsuitable habitat and one is ideal suitability. This report documents the model purpose and development as well as provides a user’s guide for the GSI toolbox.

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