Approximations related to the complete p-elliptic integrals

Lemma 2.1

(See [8, Theorem 1.25]) Let − ∞ < a < b < ∞ , f , g : [ a , b ] → R be continuous on [ a , b ] and be differentiable on ( a , b ) such that g ′ ( x ) ≠ 0 on ( a , b ) . Then both the functions [ f ( x ) − f ( a ) ] / [ g ( x ) − g ( a ) ] and [ f ( x ) − f ( b ) ] / [ g ( x ) − g ( b ) ] are (strictly) increasing (decreasing) on ( a , b ) if f ′ ( x ) / g ′ ( x ) is (strictly) increasing (decreasing) on ( a , b ) .

Lemma 2.2

(See [16, Theorem 2.1]) Suppose that the power series f ( x ) = ∑ n = 0 ∞ a n x n and g ( x ) = ∑ n = 0 ∞ b n x n have the radius of convergence r > 0 with b n > 0 for all n ∈ { 0 , 1 , 2 , … } . Let h ( x ) = f ( x ) / g ( x ) and H f , g = ( f ′ / g ′ ) g − f , then the following statements are true:

1. If the nonconstant sequence { a n / b n } ∣ n = 0 ∞ is increasing (decreasing), then h ( x ) is strictly increasing (decreasing) on ( 0 , r ) ;

2. If the nonconstant sequence { a n / b n } is increasing (decreasing) for 0 < n ≤ n 0 and decreasing (increasing) for n > n 0 , then the function h is strictly increasing (decreasing) on ( 0 , r ) if and only if H f , g ( r − ) ≥ ( ≤ ) 0 . While if H f , g ( r − ) < ( > ) 0 , then exists δ ∈ ( 0 , r ) such that h ( x ) is strictly increasing (decreasing) on ( 0 , δ ) and strictly decreasing (increasing) on ( δ , r ) .

Lemma 2.3

1. For p > 1 , the function g 1 ( r ) ≡ [ r − r ′ p arth p r ] / r p + 1 is strictly increasing and convex from ( 0 , 1 ) onto ( p / ( p + 1 ) , 1 ) .

2. For p > ( 3 + 17 ) / 4 , the function g 2 ( r ) ≡ [ 1 + ( p − 2 ) r p ] g 1 ( r ) is increasing and convex from ( 0 , 1 ) onto ( p / ( p + 1 ) , p − 1 ) .

3. For p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) , the function g 3 ( r ) ≡ [ r + ( p − 1 ) r ′ p arth p r ] / ( r r ′ ) is strictly increasing and convex from ( 0 , 1 ) onto ( p , ∞ ) .

4. For p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) , the function g 4 ( r ) ≡ ( r − r ′ p arth p r ) / [ r ′ ( arth p r − r ) ] is strictly increasing from ( 0 , 1 ) onto ( p , ∞ ) .

Proof

1. By (1.12), g 1 ( r ) can be written as follows:

   (2.4) g 1 ( r ) = 1 − r ′ p ( arth p r / r ) r p = p ∑ n = 0 ∞ r p n ( p n + 1 ) ( p n + p + 1 ) ,

   yielding the monotonicity and convexity for g 1 and the limiting value g 1 ( 0 + ) = p / ( p + 1 ) . Clearly, g 1 ( 1 − ) = 1 .

2. Clearly, g 2 ( 0 ) = p / ( p + 1 ) , g 2 ( 1 ) = p − 1 . By (2.4), g 2 can be written as follows:

   (2.5) g 2 ( r ) = p ∑ n = 0 ∞ r p n ( p n + 1 ) ( p n + p + 1 ) + ( p − 2 ) ∑ n = 0 ∞ r p ( n + 1 ) ( p n + 1 ) ( p n + p + 1 ) = p p + 1 + p ∑ n = 0 ∞ p ( p − 1 ) n + ( 2 p 2 − 3 p − 1 ) ( p n + 1 ) ( p n + p + 1 ) ( p n + 2 p + n ) r p ( n + 1 ) = p p + 1 + p ∑ n = 0 ∞ f ( n , p ) ( p n + 1 ) ( p n + p + 1 ) ( p n + 2 p + n ) r p ( n + 1 ) ,

   where f ( n , p ) = p ( p − 1 ) n + ( 2 p 2 − 3 p − 1 ) = ( n + 2 ) p 2 − ( n + 3 ) p − 1 . It is easy to see that f ( n , p ) is strictly increasing with respect to p . Then f ( n , p ) > f ( n , ( 3 + 17 ) / 4 ) = [ ( 7 + 17 ) n ] / 8 ≥ 0 . Hence, the monotonicity and convexity properties of g 2 follow from (2.5).

3. The limiting values of g 3 are clear. By (2.3) and by differentiation, we obtain

   (2.6) g 3 ′ ( r ) = [ p − p ( p − 1 ) r p − 1 arth p r ] r r ′ − [ r + ( p − 1 ) r ′ p arth p r ] ( 1 − 2 r p ) / r ′ ( p − 1 ) ( r r ′ ) 2 = r [ p r ′ p − 1 + 2 r p ] − ( p − 1 ) r ′ p arth p r [ ( p − 2 ) r p + 1 ] r 2 r ′ ( p + 1 ) = ( p − 1 ) ( r − r ′ p arth p r ) + ( 2 − p ) r p [ r + ( p − 1 ) r ′ p arth p r ] r 2 r ′ ( p + 1 ) = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ( p − 1 ) + ( 2 − p ) r p 1 + p r ′ p arth p r r − r ′ p arth p r

   = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ( p − 1 ) + ( 2 − p ) r p 1 − p + p r r − r ′ p arth p r = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ⋅ ( p − 1 ) + ( 2 − p ) ( 1 − p ) r p + p r p + 1 r − r ′ p arth p r = r p − 1 r ′ ( p + 1 ) ⋅ { ( p − 1 ) [ 1 + ( p − 2 ) r p ] g 1 ( r ) + p ( 2 − p ) } = r p − 1 r ′ ( p + 1 ) ⋅ [ ( p − 1 ) g 2 ( r ) + p ( 2 − p ) ] = r p − 1 r ′ ( p + 1 ) ⋅ G 1 ( r ) ,

   where G 1 ( r ) = ( p − 1 ) g 2 ( r ) + p ( 2 − p ) . Hence, by the part (2), G 1 is increasing on (0,1) for p > ( 3 + 17 ) / 4 . On the other hand, it is easy to verify that

   G 1 ( 0 ) = p ( − p 2 + 2 p + 1 ) p + 1 > 0

   for p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) . Consequently, by (2.6), g 3 ′ is strictly increasing on ( 0 , 1 ) with g 3 ′ ( 0 ) = 0 , and hence, the monotonicity and convexity properties of g 3 follow from part (2).

4. Let G 2 ( r ) = r / r ′ − r ′ ( p − 1 ) arth p r and G 3 ( r ) = arth p r − r . Then G 2 ( 0 ) = G 3 ( 0 ) = 0 , g 4 ( r ) = G 2 ( r ) / G 3 ( r ) and

   (2.7) G 2 ′ ( r ) / G 3 ′ ( r ) = g 3 ( r ) .

Lemma 2.4

Given p ∈ [ 2 , ∞ ) , the function h ( r ) ≡ E p / [ r ′ ( p − 1 ) K p ] + ( p − 2 ) r ′ is strictly increasing and convex from ( 0 , 1 ) onto ( p − 1 , ∞ ) .

Proof

Clearly, h ( 0 ) = p − 1 and h ( 1 ) = ∞ . Differentiation gives

where

By [23, Lemma 3.4(2),(3),(5)], for p ≥ 2 , we see that H 1 ( r ) is decreasing from (0,1) onto ( 0 , p − 1 ) . Hence, it follows from (2.8) that h ′ is the product of three positive and increasing functions. This yields the monotonicity and convexity properties of h .□

Lemma 2.5

The function j ( x ) = 1 / sin x − 1 / x − 1 / π is increasing from ( 0 , π ) onto ( − 1 / π , ∞ ) , and exists x 0 = 1.451765874910260 ⋯ ∈ ( 0 , π ) for j ( x 0 ) = 0 .

Proof

By the Taylor expansion of x / sin x ([30, Equality (2.15)]), we obtain

where B n ( n ∈ N 0 , N 0 = N ∪ { 0 } ) are the Bernoulli numbers, defined by

By differentiation,

It is easy to see that for all x ∈ ( 0 , π ) , j ′ ( x ) > 0 . Consequently, j ( x ) is increasing on ( 0 , π ) . It follows from (2.9) that the limits j ( 0 ) = − 1 / π and j ( π ) = ∞ . By the mathematical software Maple, we compute that j ( x 0 ) = 0 .□

Theorem 3.1

1. For r ∈ ( 0 , 1 ) , p ∈ [ 2 , 1 + 2 ] , the function

   f 1 ( r ) ≡ log ( 2 K p ( r ) / π p ) log ( arth p r / r )

   is strictly increasing from ( 0 , 1 ) onto ( 1 − 1 / p 2 , 1 ) . In particular, for all r ∈ ( 0 , 1 ) , p ∈ [ 2 , 1 + 2 ] , the double inequality

   (3.1) π p 2 arth p r r α 1 < K p ( r ) < π p 2 arth p r r β 1

   holds with the best possible constants α 1 = 1 − 1 / p 2 and β 1 = 1 .

2. For r ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , the function

   f 2 ( r ) = r ( 2 K p ( r ) / π p − 1 ) arth p r − r

   is strictly decreasing from ( 0 , 1 ) onto ( 2 / π p , 1 − 1 / p 2 ) . In particular, for all r ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , the double inequality

   (3.2) 1 − α 2 + α 2 arth p r r < 2 π p K p < 1 − β 2 + β 2 arth p r r

   holds with the best possible constants α 2 = 2 / π p and β 2 = 1 − 1 / p 2 .

Proof

(1) Let

Then F 1 ( 0 + ) = F 2 ( 0 + ) = F 3 ( 0 + ) = F 4 ( 0 + ) = 0 , f 1 ( r ) = F 1 ( r ) / F 2 ( r ) , and by differentiation,

where g 4 and h are, respectively, defined as in Lemmas 2.3(4) and in 2.4. It follows from (3.4), Lemmas 2.3(4) and 2.4 that the function F 3 ′ / F 4 ′ is strictly decreasing on (0,1), and is F 3 / F 4 by Lemma 2.1. Consequently, it follows from (3.3) and Lemma 2.1 that f 1 is strictly increasing on (0,1).

By Lemma 2.1 (l’Hôpital’s rule), Lemma 2.3(4), Lemma 2.4, and by (3.3)–(3.4), we obtain the limiting values:

By the monotonicity of f 1 , the double inequality (3.1) holds with α 1 = 1 − 1 / p 2 and β 1 = 1 .

It is easy to see that for all r ∈ ( 0 , 1 ) and p ∈ [ 2 , 1 + 2 ] ,