A correct proof of logarithmic coefficients for Janowski typ

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Abstract

Recently, authors [7] studied the logarithmic coefficient bounds for class of the Janowski type $(j,k)$-symmetric starlike functions $\mathcal{ST}_{[j,k]}(A,B)$ in ({\em Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.} (2022). DOI: 10.1007/s13398-022-01310-9). We pointed out that the proof of Theorem~3 in [7] is incorrect. In this article, we present the correct proof of Theorem~3. In addition, we also obtain some new results related to the logarithmic coefficient inequalities for the class

Results & Lemmas (2)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Theorem 2.3 Theorem 2.3. For, and, the logarithmic coefficients of satisfy the inequality The inequality is sharp for the function defined by (1.2).

Theorem 2.3. For $A \in \mathbb{C}$ , $-1 \leq B \leq 0$ and $A \neq B$ , the logarithmic coefficients of $f_{j,k} \in \mathcal{ST}_{[j,k]}(A,B)$ satisfy the inequality $$\sum_{n=1}^{\infty} n^2 |d_{n(j+k-1)}|^2 \le \frac{1}{4(j+k-1)^2} \frac{|A-B|^2}{1-B^2} \quad \text{for } B \ne -1.$$ The inequality is sharp for the function $K_{A,B}^{(j,k)}$ defined by (1.2).

Theorem 2.5 Theorem 2.5. Let for, and, and let. Then the logarithmic coefficients of satisfy the inequality N. L. Sharma

Theorem 2.5. Let $f_{j,k} \in \mathcal{ST}_{[j,k]}(A,B)$ for $A \in \mathbb{C}$ , $-1 \leq B \leq 0$ and $A \neq B$ , and let $t \leq 2$ . Then the logarithmic coefficients of $f_{j,k}$ satisfy the inequality $$\sum_{n=1}^{\infty} (n+1)^t |d_{n(j+k-1)}|^2 \le \left(\frac{|A-B|}{2(j+k-1)B}\right)^2 \sum_{n=1}^{\infty} \frac{(n+1)^t}{n^2} B^{2n}.$$ N. L. Sharma

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A correct proof of logarithmic coefficients for Janowski type $(j,k)$-symmetric starlike functions

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Results & Lemmas (2)

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