Let $R=k[x,y]$ be a polynomial ring in two variables over an infinite field $k$. Let $\mathfrak m=(x,y)$. Let $J$ be a homogeneous ideal whose radical is $\mathfrak m$. Consider the standard grading on $R/J$ (which is an Artinian $k$-algebra) . Let $d:=\min \{\deg (f) : 0\ne f \in J\}$ .
Then how to show that $\dim_k (R/J)_i \ge \dim_k (R/J)_{i+1}, \forall i\ge d-1$ ? (Where $(R/J)_i$ denotes the $i$-th graded part of $R/J$ )
In other words , how to show that $\mu \bigg(\dfrac {\mathfrak m^i+ J}{J}\bigg) \ge \mu \bigg(\dfrac {\mathfrak m^{i+1} + J}{J}\bigg) , \forall i\ge d-1$ ?
Here $\mu(-)$ denotes minimal number of generators
Defined $$ R_{\leq h}=\{f\in R=\mathbb{K}[x,y]\mid\deg f\leq h\},\,J_{\leq h}=\{f\in J\mid\deg f\leq h\}, $$ let $B_h$ a base of $J_{\leq h}$ as $\mathbb{K}$-vector space: $B_h$ is a finite set.
Proof. $J_{\leq h}$ is a vector subspace of $R_{\leq h}$, which has (finite) dimension $\displaystyle\sum_{i=0}^h\binom{i+1}{i}$; so $\dim_{\mathbb{K}}J_{\leq h}$ is finite. $\Box$
Remark 1. More in general, $\displaystyle\dim_{\mathbb{K}}\mathbb{K}[x_1,\dots,x_n]_{\leq h}=\sum_{i=0}^h\binom{i+n-1}{i}$. $\diamond$
Considering the canonical surjective morphism of $\mathbb{K}$-algebras $$ \pi:R\twoheadrightarrow R_{\displaystyle/J}, $$ it induces the inclusions $$ i_h:R_{\leq h\displaystyle/J_{\leq h}}\hookrightarrow R_{\displaystyle/J}; $$ let $E$ be a base of $R_{\displaystyle/J}$, then $E_{\leq h}=\{f\in E\mid\deg f\leq h\}$ is a system of generators for $R_{\leq h\displaystyle/J_{\leq h}}$, and by hypothesis $E$ and $E_{\leq h}$ are finite sets.
Dualizing, one can define an epimorphism of $\mathbb{K}$-vector spaces $$ \epsilon:R_{\displaystyle/J}\twoheadrightarrow R_{\leq h\displaystyle/J_{\leq h}} $$ where $\ker(\epsilon)=\langle f\in E\mid\deg f>h\rangle$; indeed $\mathrm{Im}(\epsilon)\cong\langle E_{\leq h}\rangle$.
In particular one has: $$ \forall h\geq d=\min\{\deg f\in\mathbb{N}_{\geq1}\mid f\in J\},\,\dim_{\mathbb{K}}R_{h+1\displaystyle/J_{\leq h+1}}\leq\dim_{\mathbb{K}}R_{h\displaystyle/J_{\leq h}}. $$ Remark 2. I didn't use any assumption neither on the undergound field $\mathbb{K}$ nor on the number $n$ of indeterminates $x_i$'s. I just used the $0$-Krull-dimensionality of $R_{\displaystyle/\sqrt{J}}$.