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T. Amdeberhan
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On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$.

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-concave if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-concave. Is the following true? If not, give a minimal added condition to secure it. $$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\geq0, \qquad \text{for all $k\geq0$}.$$$$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\leq0, \qquad \text{for all $k\geq0$}.$$

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$.

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-concave if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-concave. Is the following true? If not, give a minimal added condition to secure it. $$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\geq0, \qquad \text{for all $k\geq0$}.$$

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$.

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-concave if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-concave. Is the following true? If not, give a minimal added condition to secure it. $$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\leq0, \qquad \text{for all $k\geq0$}.$$

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T. Amdeberhan
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Does this inequality follow from doubly log-convexconcave?

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_ka_{k+2}-a_{k+1}^2$$\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-convexconcave if $\mathcal{L}a_k\geq0$ for all $k\geq0$. 

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-convexconcave if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-convexconcave. Is the following true? If not, give a minimal added condition to secure it. $$\frac{\mathcal{L}^2a_k\cdot\mathcal{L}^2a_{k+2}}{a_{k+2}\cdot a_{k+4}} -\frac{(\mathcal{L}^2a_{k+1})^2}{(a_{k+3})^2}\geq0, \qquad \text{for all $k\geq0$}.$$$$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\geq0, \qquad \text{for all $k\geq0$}.$$

Does this inequality follow from doubly log-convex?

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_ka_{k+2}-a_{k+1}^2$. Then, $(a_k)_k$ is called log-convex if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-convex if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-convex. Is the following true? If not, give a minimal added condition to secure it. $$\frac{\mathcal{L}^2a_k\cdot\mathcal{L}^2a_{k+2}}{a_{k+2}\cdot a_{k+4}} -\frac{(\mathcal{L}^2a_{k+1})^2}{(a_{k+3})^2}\geq0, \qquad \text{for all $k\geq0$}.$$

Does this inequality follow from doubly log-concave?

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$. 

One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-concave if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-concave. Is the following true? If not, give a minimal added condition to secure it. $$\frac{(\mathcal{L}a_k)^2}{a_k^2} -\frac{\mathcal{L}a_{k-1}\cdot\mathcal{L}a_{k+1}}{a_{k-2}\cdot a_{k+2}}\geq0, \qquad \text{for all $k\geq0$}.$$

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T. Amdeberhan
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  • 57
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On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_ka_{k+2}-a_{k+1}^2$. Then, $(a_k)_k$ is called log-convex if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-convex if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-convex. Is the following true? If not, give a minimal added condition to secure it. $$\frac{\mathcal{L}^2a_k\cdot\mathcal{L}^2a_{k+2}}{a_{k+2}\cdot a_{k+4}} -\frac{(\mathcal{L}^2a_{k+1})^2}{(a_{k+3})^2}\geq0, \qquad \text{for all $k\geq0$}.$$

On a sequence $(a_k)_{k\geq0}$, define the operator $\mathcal{L}a_k=a_ka_{k+2}-a_{k+1}^2$. Then, $(a_k)_k$ is called log-convex if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-convex if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-convex. Is the following true? If not, give a minimal added condition to secure it. $$\frac{\mathcal{L}^2a_k\cdot\mathcal{L}^2a_{k+2}}{a_{k+2}\cdot a_{k+4}} -\frac{(\mathcal{L}^2a_{k+1})^2}{(a_{k+3})^2}\geq0, \qquad \text{for all $k\geq0$}.$$

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_ka_{k+2}-a_{k+1}^2$. Then, $(a_k)_k$ is called log-convex if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may also consider iteratively $\mathcal{L}^2a_k=\mathcal{L}\mathcal{L}a_k$ and $(a_k)_k$ is called doubly log-convex if $\mathcal{L}a_k\geq0$ as well as $\mathcal{L}^2a_k\geq0$ for all $k\geq0$.

I would like to ask:

QUESTION. Suppose $(a_k)_k$ is doubly log-convex. Is the following true? If not, give a minimal added condition to secure it. $$\frac{\mathcal{L}^2a_k\cdot\mathcal{L}^2a_{k+2}}{a_{k+2}\cdot a_{k+4}} -\frac{(\mathcal{L}^2a_{k+1})^2}{(a_{k+3})^2}\geq0, \qquad \text{for all $k\geq0$}.$$

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T. Amdeberhan
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