Complements of nabla and delta Hardy-Copson type inequalities and their applications

In this paper the classical nabla and delta Hardy-Copson type inequalities, which are derived for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>ζ</mml:mi> <mml:mo class="MathClass-rel" stretchy="false">&gt;</mml:mo> <mml:mn>1</mml:mn><mml:mo class="MathClass-punc" stretchy="false">,</mml:mo></mml:mrow></mml:math> are complemented to the new case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>ζ</mml:mi> <mml:mo class="MathClass-rel" stretchy="false">&lt;</mml:mo> <mml:mn>0</mml:mn></mml:mrow></mml:math>. These complements have exactly the same forms as the aforementioned classical inequalities except that the exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ζ</mml:mi></mml:math> is not greater than one but it is less than zero. The obtained inequalities are not only novel but also unify the continuous and discrete cases for which the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>ζ</mml:mi> <mml:mo class="MathClass-rel" stretchy="false">&lt;</mml:mo> <mml:mn>0</mml:mn></mml:mrow></mml:math> has not been considered so far either. Moreover one of the applications of Hardy-Copson type inequalities, which is to find nonoscillation criteria for the half linear differential/dynamic/difference equations, are presented by using complementary delta Hardy-Copson type inequalities.