On the Zeroes and the Critical Points of a Solution of a Second Order Half-Linear Differential Equation

Table 5

Comparison of different lower bounds.

Reference	A	B	Constraints
Corollary 2.6	(𝑏−𝑎)^1/𝑠∫^𝑏𝑎max𝑞(𝑠),𝑎≤𝑠≤𝑥𝑑𝑥^1/𝑟,; if 𝑦^′(𝑎)=𝑦(𝑏)=0.	𝜋𝑟/2√^𝑟𝑟−1	𝑞(𝑥)>0
	(𝑏−𝑎)^1/𝑠∫^𝑏𝑎max𝑞(𝑠),𝑥≤𝑠≤𝑏𝑑𝑥^1/𝑟,; if 𝑦(𝑎)=𝑦^′(𝑏)=0.
[2, 7, 8],; [9, Lemma  4.1]	(𝑏−𝑎)^1/𝑠∫^𝑏𝑎𝑞(𝑥)𝑑𝑥^1/𝑟	1	N/A
[9, Proposition  3.3]	(𝑏−𝑎)^1/𝑠∫^𝑏𝑎𝑞(𝑥)𝑑𝑥^1/𝑟	𝜋𝑟/2	𝑞(𝑥)>0, increasing
[6]	∫^𝑏𝑎|𝑄(𝑥)|^𝑠(𝑏−𝑥)^𝑟−1𝑑𝑥 with 𝑄(𝑥)=∫^𝑥𝑎𝑞(𝑠)𝑑𝑠,; if 𝑦^′(𝑎)=𝑦(𝑏)=0.	1	N/A
	∫^𝑏𝑎|𝑄(𝑥)|^𝑠(𝑥−𝑎)^𝑟−1𝑑𝑥 with 𝑄(𝑥)=∫^𝑏𝑥𝑞(𝑠)𝑑𝑠,; if 𝑦(𝑎)=𝑦^′(𝑏)=0.