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In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.

Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and the power series \begin{equation*} A(x)=\sum_{k=0}^\infty a_kx^k\quad\text{and}\quad B(x)=\sum_{k=0}^\infty b_kx^k \end{equation*} be convergent on $(-R,R)$ for some $R>0$. If $b_k>0$ and the ratio $\frac{a_k}{b_k}$ is increasing for $k\ge0$, then the function $\frac{A(x)}{B(x)}$ is also increasing on $(0,R)$.

There are many monotonicity rules for ratios of two functions, two Laplace transforms, two functional series, and the like. See, for example, the papers [2, 3, 4] listed below.

My problem is:

About the above monotonicity rule for the ratio of two Maclaurin power series $A(x)$ and $B(x)$, is there any related monotonicity conclusion on the left semi-half interval $(-R,0)$? In other words, under what conditions about the sequence $\frac{a_k}{b_k}$ for $k\ge0$, the function $\frac{A(x)}{B(x)}$ is monotonic on $(-R,0)$?

My problem should be revised as follows:

About the above monotonicity rule for the ratio of two Maclaurin power series $A(x)$ and $B(x)$, is there any related monotonicity result on the left semi-half interval $(-R,0)$? In other words, under same conditions on the sequences $a_k$, $b_k$, and $\frac{a_k}{b_k}$ for $k\ge0$, is there any monotonicity result of the function $\frac{A(x)}{B(x)}$ on $(-R,0)$?

References

  1. M. Biernacki and J. Krzyz, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 9 (1955), 135--147 (1957).
  2. Z.-H. Yang, Y.-M. Chu, and M.-K. Wang, Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl. 428 (2015), no. 1, 587--604; available online at https://doi.org/10.1016/j.jmaa.2015.03.043.
  3. Feng Qi and Yong-Hong Yao, Increasing property and logarithmic convexity concerning Dirichlet beta function, Euler numbers, and their ratios, Hacettepe Journal of Mathematics and Statistics 52 (2023), no. 1, 17--22; available online at https://doi.org/10.15672/hujms.1099250.
  4. I. Pinelis, On L'Hospital-type rules for monotonicity, J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 40, 19 pages; available online at https://emis.de/journals/JIPAM/article657.html.
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  • $\begingroup$ @IosifPinelis Dear Professor Pinelis, you are an expert on monotonicity rules, could you please help me find a solution to this problem? Thank you very much. $\endgroup$
    – qifeng618
    Commented May 11, 2023 at 9:56
  • $\begingroup$ Of course, you can trivially do the reflection $x\leftrightarrow-x$ and at the same time change the condition $b_k>0$ to $(-1)^kb_k>0$, to get the result for $(-R,0)$ from that for $(0,R)$. However, I do not think that one can say anything useful in general for $(0,R)$ without the condition $b_k>0$. So then, one can hardly say anything useful in general for $(-R,0)$ without the condition $(-1)^b_k>0$. $\endgroup$
    – Iosif Pinelis
    Commented May 11, 2023 at 14:33
  • $\begingroup$ @IosifPinelis Thank you for your answer to my question. Could you please give a solution to my revised problem above? $\endgroup$
    – qifeng618
    Commented May 12, 2023 at 12:39
  • $\begingroup$ Unfortunately, I don't think that such results exist and would be surprised if they do exist. $\endgroup$
    – Iosif Pinelis
    Commented May 12, 2023 at 15:01

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