Certain subclasses <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{S}\mathcal{H}^{m,\ell }_{p,q}[\Phi _{i}, \Psi _{j}; \alpha , A, B]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <msubsup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>m</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">[</mo> <msub> <mi mathvariant="normal">Φ</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mi>j</mi> </msub> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{\mathcal{S}\mathcal{H}}^{m,\ell }_{p,q}[\Phi _{i}, \Psi _{j}; \alpha , A, B]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mover> <mrow> <mi mathvariant="script">S</mi> <mi mathvariant="script">H</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>m</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">[</mo> <msub> <mi mathvariant="normal">Φ</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mi>j</mi> </msub> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the function class <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{S}\mathcal{H}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of multivalent harmonic functions associated with the (p, q)-derivative operator and subordination function are introduced and investigated. Further, on the one hand, we obtain the necessary and sufficient conditions and the coefficient characterization for the class of (p, q)-Sălăgean multivalent harmonic functions for the subclasses. On the other hand, we also provide the bound estimates of the coefficient <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$b_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and distortion theorems and covering theorems as well as the closeness of convex combination for these harmonic function classes.

THE (p, q)-ANALOG OF THE SĂLĂGEAN-TYPE MULTIVALENT HARMONIC FUNCTIONS ASSOCIATED WITH SUBORDINATION

期刊论文
发表日期：

2025年4月2日

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Pinhong Long,
Jinlin Liu,
Murugusundaramoorthy Gangadharan

摘要

Certain subclasses $\mathcal{S}\mathcal{H}^{m,\ell }_{p,q}[\Phi _{i}, \Psi _{j}; \alpha , A, B]$ $S H_{p, q}^{m, ℓ} [Φ_{i}, Ψ_{j}; α, A, B]$ and $\overline{\mathcal{S}\mathcal{H}}^{m,\ell }_{p,q}[\Phi _{i}, \Psi _{j}; \alpha , A, B]$ ${\bar{S H}}_{p, q}^{m, ℓ} [Φ_{i}, Ψ_{j}; α, A, B]$ of the function class $\mathcal{S}\mathcal{H}(m)$ $S H (m)$ of multivalent harmonic functions associated with the (p, q)-derivative operator and subordination function are introduced and investigated. Further, on the one hand, we obtain the necessary and sufficient conditions and the coefficient characterization for the class of (p, q)-Sălăgean multivalent harmonic functions for the subclasses. On the other hand, we also provide the bound estimates of the coefficient $a_{n}$ $a_{n}$ and $b_{n}$ $b_{n}$ and distortion theorems and covering theorems as well as the closeness of convex combination for these harmonic function classes.