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<p>In this paper, we define and studied properties of various classes of analytic functions by using <i>q</i>-analogue of the Riemann-Liouville integral operator. By utilizing the definition of <i>q</i>-starlike functions and <i>q</i>-uniformly starlike functions, we introduce the several subclasses of univalent functions in an open unit disk. Further, we obtain some useful differential subordination results for new subclasses benefited from the <i>q</i>-analogue of the Riemann-Liouville integral operator. Moreover, we calculate necessary and sufficient conditions for existence of the given classes by using the differential subordination. We also obtain an integral representation for the functions in such new subclasses of analytic functions. We purpose some application of this study in the form of corollaries in this investigation. In addition, we generate the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((q,\delta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Jack’s Lemma for the discussed functions in a special case. Over the above, we obtain radius of starlikness of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> raised by the <i>q</i>-Riemann-integral operator.</p>

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On Special Differential Subordination Associated with Fractional q-Integral Operator

  • 期刊论文

  • 发表日期：

    2026年3月23日

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  • Ebrahim Amini,
  • Mojtaba Fardi,
  • Shrideh K. Al-Omari,
  • Areej Alshraideh

摘要

In this paper, we define and studied properties of various classes of analytic functions by using q-analogue of the Riemann-Liouville integral operator. By utilizing the definition of q-starlike functions and q-uniformly starlike functions, we introduce the several subclasses of univalent functions in an open unit disk. Further, we obtain some useful differential subordination results for new subclasses benefited from the q-analogue of the Riemann-Liouville integral operator. Moreover, we calculate necessary and sufficient conditions for existence of the given classes by using the differential subordination. We also obtain an integral representation for the functions in such new subclasses of analytic functions. We purpose some application of this study in the form of corollaries in this investigation. In addition, we generate the \((q,\delta )\) ( q , δ ) -Jack’s Lemma for the discussed functions in a special case. Over the above, we obtain radius of starlikness of order \(\beta \) β raised by the q-Riemann-integral operator.