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Improved rigorous bounds on the effective elastic and transport properties of a transversely isotropic fiber‐reinforced material composed of oriented, infinitely long, multisized circular cylinders distributed throughout a matrix are computed. Specifically, we evaluate such bounds on the effective axial shear modulus (which includes, by mathematical analogy, the transverse conductivity), effective transverse bulk modulus, and the effective transverse shear modulus. These are generally demonstrated to provide significant improvement over the Hill–Hashin bounds which incorporate only volume‐fraction information. Although the upper bounds diverge from the lower bounds when the cylinders are much stiffer than the matrix, the improved lower bounds still yield relatively accurate estimates of the effective properties. Generally, increasing the degree of polydispersivity in cylinder size increases the effective transverse conductivity (or axial shear modulus) and effective transverse bulk modulus, and decreases (slightly) the effective transverse shear modulus for cases in which the fibers are more conducting or stiffer than the matrix.