| Summary: | In this article, we make use of the <i>q</i>-binomial theorem to introduce and study two new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℵ</mo><mo>(</mo><mi>α</mi><mi>q</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℵ</mo><mo>(</mo><mi>α</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula> of meromorphic functions in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">U</mi></semantics></math></inline-formula>, that is, analytic functions in the punctured unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">U</mi><mo>∗</mo></msup><mo>=</mo><mi mathvariant="double-struck">U</mi><mo>\</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>z</mi><mo>:</mo><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mspace width="1.em"></mspace><mi>and</mi><mspace width="1.em"></mspace><mn>0</mn><mo><</mo><mfenced open="|" close="|"><mi>z</mi></mfenced><mo><</mo><mn>1</mn><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered.
|
|---|