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)
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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
Function classes studied:
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