Answer
If the barbell is accelerating upward, the force exerted by the athlete's hand must exceed the weight of the barbell, in order to provide a net upward force on the barbell. By Newton's third law, the reaction force must also be larger. This means that the barbell also pushes with greater force, downward, against the athlete's hand.
By the same reasoning, when the barbell's acceleration is downward, the barbell pushes down on the athlete's hand with a force that is smaller than its own weight.
