Function (biology)

Pure mathematics

Biology

Discrete mathematics

Monotonic function

Applied mathematics

Economics

Evolutionary biology

Organic chemistry

Upper and lower bounds

Mathematics

Elliptic integral

Mathematical analysis

Financial economics

Combinatorics

Chemistry

Convexity

Ring (chemistry)

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas

In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind $${\mathcal {K}}_a(r)$$ K a ( r ) and its approximation $$\log [1+2/(ar')]$$ log [ 1 + 2 / ( a r ′ ) ] , and also the convexity (concavity) of their difference for $$a\in (0,1/2]$$ a ∈ ( 0 , 1 / 2 ] . As an application, we give new bounds for generalized Grötzsch ring function $$\mu _a(r)$$ μ a ( r ) and a upper bound for $${\mathcal {K}}_a(r)$$ K a ( r ) .

Monotonicity and convexity involving generalized elliptic integral of the first kind