Normalized generalized Bessel function and its geometric properties
Abstract The normalization of the generalized Bessel functions U σ , r $\mathrm{U}_{\sigma,r}$ ( σ , r ∈ C ) $(\sigma,r\in \mathbb{C}\mathbbm{)}$ defined by U σ , r ( z ) = z + ∑ j = 1 ∞ ( − r ) j 4 j ( 1 ) j ( σ ) j z j + 1 $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-12-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-022-02891-0 |
| _version_ | 1811196665615351808 |
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| author | Hanaa M. Zayed Teodor Bulboacă |
| author_facet | Hanaa M. Zayed Teodor Bulboacă |
| author_sort | Hanaa M. Zayed |
| collection | DOAJ |
| description | Abstract The normalization of the generalized Bessel functions U σ , r $\mathrm{U}_{\sigma,r}$ ( σ , r ∈ C ) $(\sigma,r\in \mathbb{C}\mathbbm{)}$ defined by U σ , r ( z ) = z + ∑ j = 1 ∞ ( − r ) j 4 j ( 1 ) j ( σ ) j z j + 1 $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)^{j}}{4^{j} (1)_{j}(\sigma )_{j}}z^{j+1} \end{aligned}$$ was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that U σ , r $\mathrm{U}_{\sigma,r}$ is starlike or convex of order α ( 0 ≤ α ≤ 1 ) $(0\leq \alpha \leq 1)$ in the open unit disk. Moreover, the starlikeness and convexity of U σ , r $\mathrm{U}_{\sigma,r}$ have been considered where the leading concept of the proofs comes from the starlikeness of the power series f ( z ) = ∑ j = 1 ∞ A j z j $f(z)=\sum_{j=1}^{\infty}A_{j}z^{j}$ and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of ( z cos z ) ∗ U σ , r $(z\cos \sqrt{z} ) \ast \mathrm{U}_{\sigma,r}$ and ( sin z ) ∗ U σ , r ( z 2 ) z $(\sin z ) \ast \frac {\mathrm{U}_{\sigma,r}(z^{2})}{z}$ have been determined, where “∗” stands for the convolution between the power series. |
| first_indexed | 2024-04-12T01:02:49Z |
| format | Article |
| id | doaj.art-d6cd411551454e278536238ae3192fb8 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-12T01:02:49Z |
| publishDate | 2022-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-d6cd411551454e278536238ae3192fb82022-12-22T03:54:25ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-12-012022112610.1186/s13660-022-02891-0Normalized generalized Bessel function and its geometric propertiesHanaa M. Zayed0Teodor Bulboacă1Department of Mathematics and Computer Science, Faculty of Science, Menoufia UniversityFaculty of Mathematics and Computer Science, Babeş-Bolyai UniversityAbstract The normalization of the generalized Bessel functions U σ , r $\mathrm{U}_{\sigma,r}$ ( σ , r ∈ C ) $(\sigma,r\in \mathbb{C}\mathbbm{)}$ defined by U σ , r ( z ) = z + ∑ j = 1 ∞ ( − r ) j 4 j ( 1 ) j ( σ ) j z j + 1 $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)^{j}}{4^{j} (1)_{j}(\sigma )_{j}}z^{j+1} \end{aligned}$$ was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that U σ , r $\mathrm{U}_{\sigma,r}$ is starlike or convex of order α ( 0 ≤ α ≤ 1 ) $(0\leq \alpha \leq 1)$ in the open unit disk. Moreover, the starlikeness and convexity of U σ , r $\mathrm{U}_{\sigma,r}$ have been considered where the leading concept of the proofs comes from the starlikeness of the power series f ( z ) = ∑ j = 1 ∞ A j z j $f(z)=\sum_{j=1}^{\infty}A_{j}z^{j}$ and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of ( z cos z ) ∗ U σ , r $(z\cos \sqrt{z} ) \ast \mathrm{U}_{\sigma,r}$ and ( sin z ) ∗ U σ , r ( z 2 ) z $(\sin z ) \ast \frac {\mathrm{U}_{\sigma,r}(z^{2})}{z}$ have been determined, where “∗” stands for the convolution between the power series.https://doi.org/10.1186/s13660-022-02891-0UnivalentStarlikeConvex and close-to convex functionsConvolution (Hadamard product)Modified Bessel functionDigamma function |
| spellingShingle | Hanaa M. Zayed Teodor Bulboacă Normalized generalized Bessel function and its geometric properties Journal of Inequalities and Applications Univalent Starlike Convex and close-to convex functions Convolution (Hadamard product) Modified Bessel function Digamma function |
| title | Normalized generalized Bessel function and its geometric properties |
| title_full | Normalized generalized Bessel function and its geometric properties |
| title_fullStr | Normalized generalized Bessel function and its geometric properties |
| title_full_unstemmed | Normalized generalized Bessel function and its geometric properties |
| title_short | Normalized generalized Bessel function and its geometric properties |
| title_sort | normalized generalized bessel function and its geometric properties |
| topic | Univalent Starlike Convex and close-to convex functions Convolution (Hadamard product) Modified Bessel function Digamma function |
| url | https://doi.org/10.1186/s13660-022-02891-0 |
| work_keys_str_mv | AT hanaamzayed normalizedgeneralizedbesselfunctionanditsgeometricproperties AT teodorbulboaca normalizedgeneralizedbesselfunctionanditsgeometricproperties |