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In this paper, first, we intend to determine the relationship between the sign of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><msup><mmultiscripts><mo>Δ</mo><msub><mi>c</mi><mn>0</mn></msub><mrow></mrow></mmultiscripts><mi>β</mi></msup><mi>y</mi></mfenced><mrow><mo>(</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>β</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mo>Δ</mo><mi>y</mi></mfenced><mrow><mo>(</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, in the case we assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><msup><mmultiscripts><mo>Δ</mo><msub><mi>c</mi><mn>0</mn></msub><mrow></mrow></mmultiscripts><mi>β</mi></msup><mi>y</mi></mfenced><mrow><mo>(</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is negative. After that, by considering the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mrow><mo>ℓ</mo><mo>+</mo><mn>1</mn><mo>,</mo><mi>θ</mi></mrow></msub><mo>⊆</mo><msub><mi mathvariant="script">D</mi><mrow><mo>ℓ</mo><mo>,</mo><mi>θ</mi></mrow></msub></mrow></semantics></math></inline-formula>, which are subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, we will extend our previous result to make the relationship between the sign of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><msup><mmultiscripts><mo>Δ</mo><msub><mi>c</mi><mn>0</mn></msub><mrow></mrow></mmultiscripts><mi>β</mi></msup><mi>y</mi></mfenced><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mo>Δ</mo><mi>y</mi></mfenced><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> (the monotonicity of <i>y</i>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><msup><mmultiscripts><mo>Δ</mo><msub><mi>c</mi><mn>0</mn></msub><mrow></mrow></mmultiscripts><mi>β</mi></msup><mi>y</mi></mfenced><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> will be assumed to be negative for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>∈</mo><msubsup><mi>N</mi><mrow><msub><mi>c</mi><mn>0</mn></msub></mrow><mi>T</mi></msubsup><mo>:</mo><mo>=</mo><mrow><mo>{</mo><msub><mi>c</mi><mn>0</mn></msub><mo>,</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>,</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>T</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>∈</mo><msub><mi>N</mi><msub><mi>c</mi><mn>0</mn></msub></msub><mo>:</mo><mo>=</mo><mrow><mo>{</mo><msub><mi>c</mi><mn>0</mn></msub><mo>,</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>,</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. The last part of this work is devoted to see the possibility of information reduction regarding the monotonicity of <i>y</i> despite the non-positivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><msup><mmultiscripts><mo>Δ</mo><msub><mi>c</mi><mn>0</mn></msub><mrow></mrow></mmultiscripts><mi>β</mi></msup><mi>y</mi></mfenced><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by means of numerical simulation.