Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator
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Abstract
There are many articles in the literature dealing with the first-order and the second-order differential subordination and superordination problems for analytic functions in the unit disk, but only a few articles are dealing with the above problems in the third-order case (see, e.g., Antonino and Miller (2011) and Ponnusamy et al. (1992)). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller in (2011). Let Ω be a set in the complex plane<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow></mml:math>. Also let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi mathvariant="fraktur">p</mml:mi></mml:mrow></mml:math>be analytic in the unit disk<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi mathvariant="double-struck">U</mml:mi><mml:mo>=</mml:mo><mml:mfenced open="{" close="}" separators="|"><mml:mrow><mml:mi>z</mml:mi><mml:mo>:</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">C</mml:mi><mml:mtext> and </mml:mtext><mml:mfenced open="|" close="|" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:math>and suppose that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>ψ</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:mi mathvariant="double-struck">U</mml:mi><mml:mo>→</mml:mo><mml:mi mathvariant="double-struck">C</mml:mi></mml:math>. In this paper, we investigate the problem of determining properties of functions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi mathvariant="fraktur">p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>that satisfy the following third-order differential superordination:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>⊂</mml:mo><mml:mfenced open="{" close="}" separators="|"><mml:mrow><mml:mi>ψ</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi mathvariant="fraktur">p</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="fraktur">p</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mfenced separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="fraktur">p</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mfenced separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="fraktur">p</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi><mml:mi>′</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mfenced separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>:</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">U</mml:mi></mml:mrow></mml:mfenced></mml:math>. As applications, we derive some third-order differential subordination and superordination results for meromorphically multivalent functions, which are defined by a family of convolution operators involving the Liu-Srivastava operator. The results are obtained by considering suitable classes of admissible functions.
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