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We investigate various boundary decay estimates for $p(\cdot)$-harmonicfunctions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition($C^{1,1}$-domains) we show the boundary Harnack inequality for$p(\cdot)$-harmonic functions under the assumption that the variable exponent$p$ is a bounded Lipschitz function. The proof involves barrier functions andchaining arguments.  Moreover, we prove a Carleson type estimate for $p(\cdot)$-harmonic functionsin NTA domains in $\mathbb{R}^n$ and provide lower- and upper- growth estimatesand a doubling property for a $p(\cdot)$-harmonic measure.

The boundary Harnack inequality for variable exponent $p$-Laplacian, Carleson estimates, barrier functions and $p(\cdot)$-harmonic measures